A Q-Operator for Open Spin Chains II: Boundary Factorization

Abstract

One of the features of Baxter’s Q-operators for many closed spin chain models is that all transfer matrices arise as products of two Q-operators with shifts in the spectral parameter. In the representation-theoretical approach to Q-operators, underlying this is a factorization formula for L-operators (solutions of the Yang–Baxter equation associated to particular infinite-dimensional representations). To extend such a formalism to open spin chains, one needs a factorization identity for solutions of the reflection equation (boundary Yang–Baxter equation) associated to these representations. In the case of quantum affine $\mathfrak {sl}_2$ and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.

1 Introduction

1.1 Background and overview

Baxter first introduced his Q-operator in [Ba72, Ba73] as an auxiliary tool in the derivation of Bethe Equations for the eigenvalues of the 8-vertex model transfer matrix. The key characters in the story are the transfer matrix $\mathcal {T}(z)$ and the Q-operator $\mathcal Q(z)$. A detailed description of the essential properties of $\mathcal {T}(z)$ and $\mathcal Q(z)$ can be found in [BLZ97] (also see [VW20] and references therein); the key relation that they satisfy that leads directly to the Bethe equations is of the form

$$\begin{aligned} \mathcal T(z) \mathcal Q(z)= \alpha _+(z) \mathcal Q(q z) + \alpha _-(z) \mathcal Q(q^{-1}z), \end{aligned}$$

(1.1)

where $\alpha _\pm (z)$ are meromorphic functions and $q\in \mathbb {C}^\times$ is not a root of unity.

In the original papers of Baxter, the operator ${\mathcal {Q}}(z)$ was constructed by a brilliant but ad hoc argument; the representation-theoretic construction of ${\mathcal {Q}}(z)$ had to wait more than 20 years until the work of Bazhanov, Lukyanov and Zamolodchikov [BLZ96, BLZ97, BLZ99]. The main idea of the latter approach is to construct both $\mathcal T(z)$ and $\mathcal Q(z)$ as partial traces over different representations of the universal R-matrix ${\mathcal {R}}$ of $U_q(\widehat{\mathfrak {sl}}_2)$. The operator $\mathcal T(z)$ is a twisted trace over a two-dimensional $U_q(\widehat{\mathfrak {sl}}_2)$-representation $\Pi _z$, and $\mathcal Q(z)$ is a similarly twisted trace over an infinite-dimensional $U_q({\widehat{\mathfrak {b}}}^+)$-representation $\rho _z$, where ${\widehat{\mathfrak {b}}}^+$ is the upper Borel subalgebra of ${\widehat{\mathfrak {s}\mathfrak {l}}}_2$ (the relevant representations are defined in Sect. 4.4 of the current paper). The relation (1.1) for closed spin chains then follows immediately by considering a short exact sequence (SES) of $U_q({\widehat{\mathfrak {b}}}^+)$-representations with $\Pi _z\otimes \rho _z$ as its ‘middle’ object (cf. [FR99, Lem. 2 (2)]). For an arbitrary untwisted affine Lie algebra ${\widehat{\mathfrak {g}}}$ with upper Borel subalgebra ${\widehat{\mathfrak {b}}}^+$, the level-0 representation theory of $U_q({\widehat{\mathfrak {b}}}^+)$ was studied in [HJ12]; for the general connection with the theory of Baxter’s Q-operators see [FH15].

As well as this direct SES route to the equation, there is an alternative strategy which we refer to as the factorization approach; for closed chains see [BS90, De05, DKK06, De07, BJMST09, BLMS10]. In fact, this approach was the one taken by Bazhanov, Lukyanov and Zamolodchikov. The work that developed this formalism in language most similar to the current paper is [KT14].

In this approach, a second operator $\overline{ \mathcal Q}(z)$ with similar properties to $\mathcal Q(z)$ is introduced as a trace of ${\mathcal {R}}$ over another infinite-dimensional representation ${{\bar{\varrho }}}_z$ of $U_q({\widehat{\mathfrak {b}}}^+)$. The affinized version $\upsilon _z$ of the $U_q(\mathfrak {s}\mathfrak {l}_2)$-Verma module is also considered as well as another infinite-dimensional filtered $U_q({\widehat{\mathfrak {b}}}^+)$-module $\phi _z$; these two representations depend on a complex parameter $\mu$. The key connection between all representations is given by Theorem 4.4, which expresses the fact that particular pairwise tensor products are isomorphic as $U_q({\widehat{\mathfrak {b}}}^+)$-modules by means of an explicit intertwiner ${\mathcal {O}}$ (defined in Sect. 4.5 of the current paper). At the level of the L-operators this implies

$$\begin{aligned} {\mathcal {O}}_{12} {\mathcal {L}}_{\varrho }( q^{-\mu /2} z)_{13} {\mathcal {L}}_{{{\bar{\varrho }}}}(q^{\mu /2} z)_{23} = {\mathcal {L}}_{\upsilon }(z)_{13} {\mathcal {L}}_{\phi }(z)_{23} {\mathcal {O}}_{12}, \end{aligned}$$

(1.2)

(see Theorem 5.2 of the current paper), which is referred to as factorization of the Verma module L-operator ${\mathcal {L}}_\upsilon (z)$ in terms of the L-operators ${\mathcal {L}}_{\varrho }(z)$ and ${\mathcal {L}}_{{{\bar{\varrho }}}}(z)$ which are used to define $\mathcal Q(z)$, $\overline{\mathcal Q}(z)$ (the transfer matrix corresponding to the additional operator ${\mathcal {L}}_{\phi } (z)$ is trivial).

Defining $\mathcal T_{\mu }(z)$ to be the transfer matrix that is the trace over the $\mu$-dependent representation $\upsilon _{z}$ of ${\mathcal {R}}$ in the first space, Theorem 5.2 yields a relation of the following form:

$$\begin{aligned} \mathcal T_{\mu }(z) \, \propto \, \mathcal Q(zq^{-\mu /2}) \overline{\mathcal Q}(zq^{\mu /2}). \end{aligned}$$

(1.3)

The SES associated with $\upsilon _{z}$ in the case $\mu$ is an integer then leads to the key relation (1.1).

1.2 Present work

In the current work we are interested in an analogue of (1.2) for open chains, setting out an approach to Q-operators which complements the SES approach of [VW20].

The problem of Q-operators for open XXZ chains with diagonal boundaries was discussed in [BT18] and in [Ts21]. The XXX version of this problem was solved already in [FS15]. Earlier, Baxter TQ-relations with more general boundary conditions were found in [YNZ06] (XXZ) and [YZ06] (XYZ) by spin-j transfer matrix asymptotics.

Our main result is the following analogue of Theorem 5.2, which we call the boundary factorization identity. Its existence answers in the positive a question raised in [BT18, Sec. 5]:

$$\begin{aligned} {\mathcal {K}}_\upsilon (z)_1 {\mathcal {R}}_{\upsilon \phi }(z^2) {\mathcal {K}}_\phi (z)_2 \,{\mathcal {O}}= {\mathcal {O}}{\mathcal {K}}_\varrho (q^{-\mu /2}z)_1 {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z^2) {\mathcal {K}}_{{\bar{\varrho }}}(q^{\mu /2} z )_2 \end{aligned}$$

(1.4)

where z is a formal parameter (which can be specialized to generic complex numbers). The precise statement is given in Theorem 8.1. This formula involves the actions of the universal R-matrix of $U_q({\widehat{\mathfrak {s}\mathfrak {l}}}_2)$ in tensor products of the various infinite-dimensional representations introduced. In addition, the various K-operators are diagonal solutions of reflection equations (boundary Yang–Baxter equations) [Ch84, Sk88]. They arise as actions of the universal K-matrix associated to the augmented q-Onsager algebra, a particular coideal subalgebra of $U_q({\widehat{\mathfrak {s}\mathfrak {l}}}_2)$, which featured also in e.g. [BB13, RSV15, BT18, VW20]. More precisely, diagonal solutions of the reflection equation with a free parameter, considered by Sklyanin in his 2-boundary version of the algebraic Bethe ansatz in [Sk88], are intertwiners for this algebra.

Equation (1.4) has a natural diagrammatic formulation, see Sect. 8. In a subsequent paper the authors will explain how (1.4) yields relations analogous to (1.3) and hence (1.1) for open chains.

The proof of (1.4) and of the well-definedness of the various K-operators is an application of the universal K-matrix formalism developed in [AV22a, AV22b] which is built on the earlier works [BW18, BK19]. More precisely, it relies on an extension of the theory of K-matrices for finite-dimensional representations of quantum affine algebras in [AV22b] to level-0 representations of $U_q({\widehat{\mathfrak {b}}}^+)$, which we discuss in Sect. 3. The key point is that, for the special case of the augmented q-Onsager algebra, there exists a universal element ${\mathcal {K}}$, centralizing the augmented q-Onsager algebra up to a twist, with three desirable properties.

• (i) The element ${\mathcal {K}}$ lies in (a completion of) the Borel subalgebra $U_q({\widehat{\mathfrak {b}}}^+)$, so that the resulting family of linear maps is itself compatible with $U_q({\widehat{\mathfrak {b}}}^+)$-intertwiners (which play an essential role in the algebraic theory of Baxter Q-operators).

• (ii) The coproduct of ${\mathcal {K}}$ is of a particularly simple form, which is relevant for the proof of the boundary factorization identity.

• (iii) The linear operators accomplishing the action of ${\mathcal {K}}$ in level-0 representations satisfy the untwisted reflection equation.

Thus we obtain the factorization identity (1.4) as a natural consequence of the representation theory of $U_q({\widehat{\mathfrak {s}\mathfrak {l}}}_2)$. The main benefit of this universal approach is that laborious linear-algebraic computations are avoided; in particular, we not even need explicit expressions for the various factors. Nevertheless, we do provide these explicit expressions, as we expect them to be useful in further work in this direction. We also give an alternative computational proof of (1.4), to illustrate the power of the universal approach.

This is a ‘boundary counterpart’ to the level-0 theory of the universal R-matrix, which we also include for reference. We do this in Sect. 2, staying close to the original work by Drinfeld and Jimbo [Dr85, Dr86, Ji86a, Ji86b]. In particular, Theorem 2.4 states that the grading-shifted universal R-matrix has a well-defined action as a linear-operator-valued formal power series on any tensor product of level-0 representations of $U_q({\widehat{\mathfrak {b}}}^+)$ and $U_q({\widehat{\mathfrak {b}}}^-)$ (including finite-dimensional representations). Often this well-definedness is tacitly assumed, see e.g. [VW20, Sec. 2.3]. Alternatively, it follows from the Khoroshkin-Tolstoy factorization [KT92] of the universal R-matrix, see also [BGKNR10, BGKNR13, BGKNR14]; however we are unaware of such a factorization for the universal K-matrix.

1.3 Outline

In Sect. 2 we study the action of the universal R-matrix of quantum affine $\mathfrak {s}\mathfrak {l}_2$ on tensor products of level-0 representations of Borel subalgebras. Section 3 is a ‘boundary counterpart’ to Sect. 2, where we consider the augmented q-Onsager algebra. We show that its (semi-)standard universal K-matrix, see [AV22a, AV22b], has a well-defined action on level-0 representations of $U_q({\widehat{\mathfrak {b}}}^+)$, see Theorem 3.6, and, with a simple correction, satisfies the above three desirable properties.

In Sect. 4 we discuss the relevant representations of $U_q({\widehat{\mathfrak {b}}}^+)$ in terms of (an extension of) the q-oscillator algebra, as well as the $U_q({\widehat{\mathfrak {b}}}^+)$-intertwiner ${\mathcal {O}}$. Various solutions of Yang–Baxter equations are obtained in Sect. 5 as actions of the universal R-matrix in tensor products of Borel representations. Similarly, in Sect. 6 we introduce solutions of the reflection equation as actions of the universal K-matrix in Borel representations.

We revisit the SES approach to Baxter’s Q-operators for the open XXZ spin chain in light of the universal K-matrix formalism in Sect. 7. Next, in Sect. 8 we give a diagrammatic motivation of the boundary factorization identity (1.4) for the open XXZ spin chain, and provide a short proof using the level-0 theory developed in Sect. 3. Finally, in Sect. 9 we summarize the main results and point out future work.

Some supplementary material is given in appendices. Namely, Appendix A provides some background material on deformed Pochhammer symbols and exponentials. Moreover, Appendix B contains derivations of the explicit expressions of the two R-operators appearing in (1.4). In Appendix C we provide a computational alternative proof of the boundary factorization identity (1.4), relying on the explicit expressions of all involved factors. The key tool of this proof is provided by Lemma C.1, which consists in two product formulas involving deformed Pochhammer symbols and exponentials. We emphasize that the main text and its results do not rely on Appendices B and C.

2 Quantum Affine $\mathfrak {s}\mathfrak {l}_2$ and its Universal R-Matrix

In this section we study the action of the universal R-matrix of the quasitriangular Hopf algebra quantum affine $\mathfrak {s}\mathfrak {l}_2$ on tensor products of level-0 representations (including infinite-dimensional representations) of the Borel subalgebras. We give a basic survey of the algebras involved, the representations and the quasitriangular structure and show that the universal R-matrix has a well-defined action on tensor products of all level-0 representations of the Borel subalgebras.

2.1 General overview of finite-dimensional R-matrix theory

To formulate a quantum integrable system in terms of a transfer matrix built out of R-matrices, one needs finite-dimensional representations of a suitable quasitriangular Hopf algebra. To get trigonometric R-matrices, one can proceed as follows.

Let $\mathfrak {g}$ be a finite-dimensional simple Lie algebra and note that the untwisted loop algebra $L\mathfrak {g}= \mathfrak {g}\otimes \mathbb {C}[t,t^{-1}]$ has a central extension ${\widehat{\mathfrak {g}}} = L\mathfrak {g}\oplus \mathbb {C}c$. In turn, this can be extended to $\widetilde{\mathfrak {g}} = {\widehat{\mathfrak {g}}} \oplus \mathbb {C}d$ where d satisfies $[d,\cdot ] = t \frac{\textsf{d}}{{\textsf{d}}t}$. For a fixed Cartan subalgebra $\mathfrak {h}\subset \mathfrak {g}$ we define

$$\begin{aligned} {\widehat{\mathfrak {h}}}:= \mathfrak {h}\oplus \mathbb {C}c, \qquad \widetilde{\mathfrak {h}}:= {\widehat{\mathfrak {h}}} \oplus \mathbb {C}d. \end{aligned}$$

The Lie algebra $\widetilde{\mathfrak {g}}$ is a Kac–Moody algebra and hence has a non-degenerate bilinear form $(\cdot ,\cdot )$, which restricts to a non-degenerate bilinear form on $\widetilde{\mathfrak {h}}$. See e.g. [Ka90] for more detail.

The universal enveloping algebras $U({\widehat{\mathfrak {g}}})$ and $U(\widetilde{\mathfrak {g}})$ can be q-deformed, yielding non-cocommutative Hopf algebras (Drinfeld-Jimbo quantum groups) $U_q({\widehat{\mathfrak {g}}})$ and $U_q(\widetilde{\mathfrak {g}})$, see e.g. [Dr85, Dr86, Ji86a, KT92, Lu94]. The nondegenerate bilinear form $(\cdot ,\cdot )$ lifts to $U_q(\widetilde{\mathfrak {g}})$ inducing a pairing between the q-deformed Borel subalgebras and hence a quasitriangular structure. On the other hand, the subalgebra $U_q({\widehat{\mathfrak {g}}})$ has a rich finite-dimensional representation theory, see e.g. [CP94, CP95, Ch02, HJ12]. The grading-shifted universal R-matrix has a well-defined action on tensor products of finite-dimensional representations of $U_q({\widehat{\mathfrak {g}}})$ as a formal power series, see e.g. [Dr86, FR92, KS95, EM03, He19]). We now discuss the natural extension of this theory to level-0 representations of Borel subalgebras, including various infinite-dimensional representations. We will restrict to the case $\mathfrak {g}= \mathfrak {s}\mathfrak {l}_2$ (but the theory generalizes to any quantum untwisted affine algebra).

2.2 Quantum affine $\mathfrak {s}\mathfrak {l}_2$

Denoting the canonical Cartan generator of $\mathfrak {s}\mathfrak {l}_2$ by $h_1$, ${\widehat{\mathfrak {h}}}$ is spanned by $h_0 = c-h_1$ and $h_1$. The bilinear form on $\widetilde{\mathfrak {h}}$ is defined by

$$\begin{aligned} (h_0,h_0)=(h_1,h_1)=-(h_0,h_1)=2, \qquad (h_0,d)=1, \qquad (h_1,d)=(d,d)=0. \end{aligned}$$

Fix $\epsilon \in \mathbb {C}$ such that $q = \exp (\epsilon )$ is not a root of unity. For all $\mu \in \mathbb {C}$ we will denote $\exp (\epsilon \mu )$ by $q^\mu$. First, we define $U_q(\mathfrak {g})$ as the algebra generated over $\mathbb {C}$ by e, f and invertible k subject to the relations

$$\begin{aligned} k e = q^2 e k, \qquad k f = q^{-2} f k, \qquad [e,f] = \frac{k-k^{-1}}{q-q^{-1}}. \end{aligned}$$

(2.1)

The following assignments determine a coproduct $\Delta : U_q(\mathfrak {g}) \rightarrow U_q(\mathfrak {g}) \otimes U_q(\mathfrak {g})$:

$$\begin{aligned} \Delta (e) = e \otimes 1 + k \otimes e, \qquad \Delta (f) = f \otimes k^{-1} + 1 \otimes f, \qquad \Delta (k^{\pm 1}) = k^{\pm 1} \otimes k^{\pm 1}.\nonumber \\ \end{aligned}$$

(2.2)

It uniquely extends to a Hopf algebra structure on $U_q(\mathfrak {g})$. Now the main algebra of interest, $U_q({\widehat{\mathfrak {g}}})$, arises as follows.

Definition 2.1

(Quantum affine $\mathfrak {s}\mathfrak {l}_2$ ). We denote by $U_q({\widehat{\mathfrak {g}}})$ the Hopf algebra generated by two triples $\{ e_i, f_i, k_i \}$ ($i \in \{0,1\}$), such that:

• (i) the following assignments for $i \in \{0,1\}$ define Hopf algebra embeddings from $U_q(\mathfrak {g})$ to $U_q({\widehat{\mathfrak {g}}})$:

  $$\begin{aligned} e \mapsto e_i, \qquad f \mapsto f_i, \qquad k \mapsto k_i; \end{aligned}$$

  (2.3)

• (ii) the following cross relations are satisfied:

  $$\begin{aligned}&\displaystyle k_i k_j = k_j k_i, \qquad k_i e_{j} = q^{-2} e_{j} k_i, \qquad k_i f_{j} = q^2 f_{j} k_i, \qquad [e_i,f_{j}] = 0, \end{aligned}$$

  (2.4)
  $$\begin{aligned}&\displaystyle [e_i,[e_i,[e_i,e_{j}]_{q^2}]_1]_{q^{-2}} = [f_i,[f_i,[f_i,f_{j}]_{q^2}]_1]_{q^{-2}} = 0, \end{aligned}$$

  (2.5)

  for $i \ne j$, where we have introduced the notation $[x,y]_p:= xy-pyx$.

Consider the affine Cartan subalgebra ${\widehat{\mathfrak {h}}} = \mathbb {C}h_0 \oplus \mathbb {C}h_1$. Note that its q-deformation $U_q({\widehat{\mathfrak {h}}}) = \langle k_0^{\pm 1}, k_1^{\pm 1} \rangle$ is isomorphic to the group algebra of the affine co-root lattice

$$\begin{aligned} {\widehat{Q}}^\vee = \mathbb {Z}h_0 + \mathbb {Z}h_1 \subset {\widehat{\mathfrak {h}}}. \end{aligned}$$

(2.6)

The nontrivial diagram automorphism $\Phi$ of the affine Dynkin diagram, i.e. the nontrivial permutation of the index set $\{0,1\}$, lifts to a linear automorphism $\Phi$ of ${\widehat{\mathfrak {h}}}$ which preserves the lattice ${\widehat{Q}}^\vee$. Accordingly, it also lifts to an involutive Hopf algebra automorphism of $U_q({\widehat{\mathfrak {g}}})$, also denoted $\Phi$, via the assignments

$$\begin{aligned} \Phi (e_i) = e_{\Phi (i)}, \qquad \Phi (f_i) = f_{\Phi (i)}, \qquad \Phi (k_i^{\pm 1}) = k_{\Phi (i)}^{\mp 1} \qquad \text {for } i \in \{0,1\}. \end{aligned}$$

(2.7)

2.3 Quantized Kac–Moody algebra

To define the quantized Kac–Moody algebra $U_q(\widetilde{\mathfrak {g}})$, one chooses an extension $\widetilde{Q}^\vee$ of ${\widehat{Q}}^\vee$ (a lattice of rank 3 contained in $\widetilde{\mathfrak {h}}$) preserved by $\Phi$.

Remark 2.2

The standard extension of the affine co-root lattice $\mathbb {Z}h_0 + \mathbb {Z}h_1 + \mathbb {Z}d$ is not so convenient for us, mainly in view of the construction of the universal K-matrix in Sect. 3.3. Namely, extensions of $\Phi$ to $\widetilde{\mathfrak {h}}$ which are compatible with the bilinear form on $\widetilde{\mathfrak {h}}$ do not preserve this lattice, see also [Ko14, Sec. 2.6] and [AV22a, Sec. 3.14].

The most convenient choice is to use the principal grading and set

$$\begin{aligned} d_{\textsf {pr}}:= -\frac{1}{8} {{h}_0} + \frac{3}{8} {{h}_1} + 2 {d} \in \mathfrak {h}, \end{aligned}$$

(2.8)

so that

$$\begin{aligned} (d_{\textsf {pr}},h_0)=(d_{\textsf {pr}},h_1)=1, \qquad (d_{\textsf {pr}},d_{\textsf {pr}})=0. \end{aligned}$$

Now we set $\Phi (d_{\textsf {pr}})=d_{\textsf {pr}}$ and obtain a linear automorphism $\Phi$ of $\widetilde{\mathfrak {h}}$ preserving the lattice

$$\begin{aligned} \widetilde{Q}^\vee := \mathbb {Z}h_0 + \mathbb {Z}h_1 + \mathbb {Z}d_\textsf{pr}. \end{aligned}$$

The corresponding dual map on $\widetilde{\mathfrak {h}}^*$, also denoted by $\Phi$, preserves the extended affine weight lattice

$$\begin{aligned} \widetilde{P} = \{ \lambda \in \widetilde{\mathfrak {h}}^* \, | \, \lambda (\widetilde{Q}^\vee ) \subseteq \mathbb {Z}\}. \end{aligned}$$

(2.9)

Accordingly, we define $U_q(\widetilde{\mathfrak {g}})$ as the Hopf algebra obtained by extending $U_q({\widehat{\mathfrak {g}}})$ by a group-like element¹g satisfying

$$\begin{aligned} g e_i = q e_i g, \qquad g f_i = q^{-1} f_i g, \qquad g k_i = k_i g. \end{aligned}$$

(2.10)

Hence, the assignment $\Phi (g)=g$ together with (2.7) defines an involutive Hopf algebra automorphism of $U_q({\widetilde{\mathfrak {g}}})$.

2.4 Co-opposite Hopf algebra structure

For any $\mathbb {C}$-algebra A, denote by $\sigma$ the algebra automorphism of $A \otimes A$ which sends $a \otimes a'$ to $a' \otimes a$ for all $a,a' \in A$. If $X \in A \otimes A$ we will also write $X_{21}$ for $\sigma (X)$.

If A is a bialgebra with coproduct $\Delta$, the co-opposite bialgebra, denoted $A^\textsf{cop}$, is the bialgebra with the same underlying algebra structure and counit as A but with $\Delta$ replaced by

$$\begin{aligned} \Delta ^\textsf{op}:= \sigma \circ \Delta \end{aligned}$$

(2.11)

(if A is a Hopf algebra with invertible antipode S, then $A^\textsf{cop}$ is also a Hopf algebra with antipode $S^{-1}$). The assignments

$$\begin{aligned} \omega (e_i) = f_i, \qquad \omega (f_i) = e_i, \qquad \omega (k_i^{\pm 1}) = k_i^{\mp 1} \qquad \text {for } i \in \{0,1\}, \qquad \omega (g) = g^{-1}\nonumber \\ \end{aligned}$$

(2.12)

define a bialgebra isomorphism from $U_q({\widetilde{\mathfrak {g}}})$ to $U_q({\widetilde{\mathfrak {g}}})^\textsf{cop}$ (in particular, $(\omega \otimes \omega ) \circ \Delta = \Delta ^\textsf{op} \circ \omega$) which commutes with $\Phi$.

2.5 Weight modules

We review some basic representation-theoretic notions for $U_q({\widetilde{\mathfrak {g}}})$ by means of which its universal R-matrix can be described. Consider the commutative subalgebra

$$\begin{aligned} U_q({\widetilde{\mathfrak {h}}}) = \langle k_0^{\pm 1}, k_1^{\pm 1}, g^{\pm 1} \rangle \subset U_q({\widetilde{\mathfrak {g}}}). \end{aligned}$$

(2.13)

Call a $U_q({\widetilde{\mathfrak {g}}})$-module M a $U_q({\widetilde{\mathfrak {h}}})$-weight module if

$$\begin{aligned} M = \bigoplus _{\lambda \in {\widetilde{P}}} M_\lambda , \qquad M_\lambda = \{ m \in M \, | \, k_i \cdot m = q^{\lambda (h_i)} m \text { for } i \in \{0,1\}, \, g \cdot m = q^{\lambda (d_\textsf{pr})} m \}. \end{aligned}$$

Elements of $M_\lambda$ are said to have weight $\lambda$. The adjoint action of $U_q({\widetilde{\mathfrak {h}}})$ (with its generators acting by conjugation) endows $U_q({\widetilde{\mathfrak {g}}})$ itself with a $U_q({\widetilde{\mathfrak {h}}})$-weight module structure, with elements of $U_q({\widetilde{\mathfrak {h}}})$ of weight 0. More precisely, the weights of $U_q({\widetilde{\mathfrak {g}}})$ are given by the affine root lattice

$$\begin{aligned} {\widehat{Q}}:= \mathbb {Z}\alpha _0 + \mathbb {Z}\alpha _1 \subset {\widetilde{P}} \end{aligned}$$

($e_i$ has weight $\alpha _i$, $f_i$ has weight $-\alpha _i$). The adjoint action of $U_q({\widetilde{\mathfrak {h}}})$ preserves the subalgebras

$$\begin{aligned} U_q({\widehat{\mathfrak {n}}}^+)= \langle e_0,e_1 \rangle , \qquad U_q({\widehat{\mathfrak {n}}}^-)= \langle f_0,f_1 \rangle \end{aligned}$$

(2.14)

and the corresponding weights are given by the monoids $\pm {\widehat{Q}}^+$ respectively, where ${\widehat{Q}}^+:= \mathbb {Z}_{\geqslant 0} \alpha _0 + \mathbb {Z}_{\geqslant 0} \alpha _1$.

2.6 Quasitriangularity

The universal R-matrix for $U_q({\widetilde{\mathfrak {g}}})$ is an element of a completion of $U_q({\widetilde{\mathfrak {g}}}) \otimes U_q({\widetilde{\mathfrak {g}}})$ satisfying

$$\begin{aligned}&\displaystyle {\mathcal {R}}\Delta (u)={} \Delta ^{\textsf {op}}(u) {\mathcal {R}}\qquad \text{ for } \text{ all } u \in U_q({\widetilde{\mathfrak {g}}}), \end{aligned}$$

(2.15)

$$\begin{aligned}&\displaystyle (\Delta \otimes \textsf {id})({\mathcal {R}})= {} {\mathcal {R}}_{13} {\mathcal {R}}_{23}, \qquad \qquad (\textsf {id}\otimes \Delta )({\mathcal {R}}) = {\mathcal {R}}_{13} {\mathcal {R}}_{12} \end{aligned}$$

(2.16)

and hence

$$\begin{aligned} {\mathcal {R}}_{12} {\mathcal {R}}_{13} {\mathcal {R}}_{23} = {\mathcal {R}}_{23} {\mathcal {R}}_{13} {\mathcal {R}}_{12}. \end{aligned}$$

(2.17)

Consider the quantum analogues of the Borel subalgebras, which are the Hopf subalgebras

$$\begin{aligned} U_q({\widetilde{\mathfrak {b}}}^\pm ) = \langle U_q({\widetilde{\mathfrak {h}}}), U_q({\widehat{\mathfrak {n}}}^\pm ) \rangle . \end{aligned}$$

The element $\mathcal {R}$ arises as the canonical element of the bialgebra pairing between $U_q({\widetilde{\mathfrak {b}}}^+)$ and the algebra $U_q(\widetilde{\mathfrak {b}}^-)^\textsf{op}$ (the bialgebra isomorphic as a coalgebra to $U_q(\widetilde{\mathfrak {b}}^-)$ but with the opposite multiplication), see [Dr85, Lu94]. In particular, ${\mathcal {R}}$ lies in a completion of $U_q({\widetilde{\mathfrak {b}}}^+) \otimes U_q({\widetilde{\mathfrak {b}}}^-)$. Further, invariance properties of the bialgebra pairing imply

$$\begin{aligned} (\omega \otimes \omega )({\mathcal {R}})&= {\mathcal {R}}_{21}, \end{aligned}$$

(2.18)

$$\begin{aligned} (\Phi \otimes \Phi )({\mathcal {R}})&= {\mathcal {R}}. \end{aligned}$$

(2.19)

Also, this pairing has a non-degenerate restriction to $U_q({\widehat{\mathfrak {n}}}^+)_\lambda \times U_q({\widehat{\mathfrak {n}}}^-)_{-\lambda }$ for all $\lambda \in {\widehat{Q}}^+$; denote the canonical element of this restricted pairing by $\Theta _\lambda$. With our choice of the coproduct we have

$$\begin{aligned} {\mathcal {R}}= \Theta ^{-1} \cdot \kappa ^{-1}, \qquad \Theta = \sum _{\lambda \in {\widehat{Q}}^+} \Theta _\lambda , \qquad \end{aligned}$$

(2.20)

A priori, $\Theta$ acts naturally on $U_q({\widetilde{\mathfrak {g}}})$-modules with a locally finite action of $U_q({\widehat{\mathfrak {n}}}^+)$ or $U_q({\widehat{\mathfrak {n}}}^-)$. We briefly explain one possible definition² of the element $\kappa$. The non-degenerate bilinear form $(\cdot ,\cdot )$ on ${\widetilde{\mathfrak {h}}}$ induces one on ${\widetilde{\mathfrak {h}}}^*$, which we denote by the same symbol. If $M,M'$ are $U_q({\widetilde{\mathfrak {h}}})$-weight modules we define a linear map $\kappa _M: M \otimes M' \rightarrow M \otimes M'$ by stipulating that it acts on $M_\lambda \otimes M'_{\lambda '}$ ($\lambda ,\lambda ' \in {\widetilde{P}}$) as multiplication by $q^{(\lambda ,\lambda ')}$. The family of these maps $\kappa _M$, where M runs through all $U_q({\widetilde{\mathfrak {h}}})$-weight modules, is compatible with $U_q({\widetilde{\mathfrak {h}}})$-intertwiners. Hence it gives rise to a well-defined weight-0 element $\kappa$ of the corresponding completion of $U_q({\widetilde{\mathfrak {g}}}) \otimes U_q({\widetilde{\mathfrak {g}}})$ which we call here weight completion. Similarly, we will define weight-0 elements of the weight completion of $U_q({\widetilde{\mathfrak {g}}})$ itself using functions from $\widetilde{P}$ to $\mathbb {C}$. See also [AV22a, Sec. 4.8] for more detail.

2.7 Level-0 representations

Consider the following subalgebras of $U_q({\widehat{\mathfrak {g}}})$:

$$\begin{aligned} U_q({\widehat{\mathfrak {b}}}^\pm ) = \langle U_q({\widehat{\mathfrak {h}}}), U_q({\widehat{\mathfrak {n}}}^\pm ) \rangle = U_q({\widetilde{\mathfrak {b}}}^\pm ) \cap U_q({\widehat{\mathfrak {g}}}). \end{aligned}$$

(2.21)

Then $U_q({\widehat{\mathfrak {b}}}^+)$ is isomorphic to the algebra with generators $e_i$, $k_i$ ($i \in \{0,1\}$) subject to those relations in Definition 2.1 which do not involve the $f_i$ (the proof of e.g. [Ja96, Thm. 4.21] applies). We say that a $U_q({\widehat{\mathfrak {b}}}^+)$-module V is level-0 if it decomposes as

$$\begin{aligned} V = \bigoplus _{c \in \mathbb {C}^\times } V(c), \qquad V(c) = \{ v \in V \, | \, k_0 \cdot v = c^{-1} v, \quad k_1 \cdot v = c v \} \end{aligned}$$

(2.22)

with each weight subspace V(c) finite-dimensional. Note that the class of finitely generated level-0 modules is closed under tensor products. By the $U_q({\widehat{\mathfrak {g}}})$-relations we have $e_0 \cdot V(c) \subseteq V(q^{-2}c)$, $e_1 \cdot V(c) \subseteq V(q^2c)$. It is convenient to call the subset $\{ c \in \mathbb {C}^\times \, | \, \dim (V(c)) \ne 0 \}$ the support of V. If V is a finite-dimensional $U_q({\widehat{\mathfrak {g}}})$-module then it is level-0 with support contained in $\pm q^\mathbb {Z}$, see e.g. [CP95, Prop. 12.2.3].

Remark 2.3

The $U_q({\widehat{\mathfrak {g}}})$-action on a nontrivial finite-dimensional module does not extend to a $U_q({\widetilde{\mathfrak {g}}})$-action. By [HJ12, Prop. 3.5], this is a special case of the following observation. If V is an irreducible level-0 $U_q({\widehat{\mathfrak {b}}}^+)$-module with $\dim (V)>1$, then the $U_q({\widehat{\mathfrak {b}}}^+)$-action does not extend to a $U_q(\widetilde{\mathfrak {b}}^+)$-action. To see this, choose distinct $c,c' \in \mathbb {C}^\times$ in the support of V. By irreducibility, for any nonzero $v \in V(c)$, $v' \in V(c')$ there exist $x,x' \in U_q({\widehat{\mathfrak {b}}}^+)$ such that $x \cdot v = v'$, $x' \cdot v' = v$. Without loss of generality, we may assume both x and $x'$ have no term in $U_q({\widehat{\mathfrak {h}}})$. Hence $x'x$ is not a scalar. For any nonzero $v \in V(c)$, since the action of g preserves V(c), applying g to $(x'x) \cdot v = v$ now yields a contradiction with (2.10).

Analogous definitions and comments can be made for $U_q({\widehat{\mathfrak {b}}}^-)$-modules.

2.8 The action of ${\mathcal {R}}$ on tensor products of level-0 modules

We wish to connect the quasitriangular structure of $U_q({\widetilde{\mathfrak {g}}})$ with the level-0 representation theory of $U_q({\widehat{\mathfrak {g}}})$, i.e. let the universal R-matrix of $U_q({\widetilde{\mathfrak {g}}})$ act on tensor products of level-0 modules. To do this, we follow the ideas from [Dr86, Sec. 13] (also see [FR92, Sec. 4], [He19, Sec. 1]). If we write the action of $k_1$ on an arbitrary level-0 module V as $\exp (\epsilon H_V)$, then note that the factor $\kappa$ naturally acts on tensor products $V \otimes V'$ of level-0 modules as $\exp (\epsilon H_{V} \otimes H_{V'}/2)$.

To let $\Theta$ act on such tensor products, we extend the field of scalars $\mathbb {C}$ to the Laurent polynomial ring $\mathbb {C}[z,z^{-1}]$, where z is a formal parameter. The action of $\Theta$ is particularly well-behaved if we use the principal grading. That is, we define a Hopf algebra automorphism $\Sigma _z$ of $U_q({\widetilde{\mathfrak {g}}})[z,z^{-1}]$ such that

$$\begin{aligned} \Sigma _z(e_i) = z e_i, \qquad \Sigma _z(f_i) = z^{-1} f_i, \qquad \Sigma _z|_{U_q({\widetilde{\mathfrak {h}}})} = \textsf{id}. \end{aligned}$$

(2.23)

Straightforwardly one sees that

$$\begin{aligned} \omega \circ \Sigma _z&= \Sigma _{z^{-1}} \circ \omega , \end{aligned}$$

(2.24)

$$\begin{aligned} \Phi \circ \Sigma _z&= \Sigma _z \circ \Phi . \end{aligned}$$

(2.25)

Let the height function ${\textsf {ht}}: {\widehat{Q}} \rightarrow \mathbb {Z}$ be defined by ${\textsf {ht}}(m_0 \alpha _0 + m_1 \alpha _1)=m_0+m_1$ for all $m_0,m_1 \in \mathbb {Z}$ and note that the number of elements of ${\widehat{Q}}^+$ of given height is finite. The key observation is that

$$\begin{aligned} (\Sigma _z \otimes {\textsf {id}})(\Theta ) = ({\textsf {id}}\otimes \Sigma _{z^{-1}})(\Theta ) = \sum _{r \geqslant 0} z^r \sum _{\lambda \in {\widehat{Q}}^+, \, {\textsf {ht}}(\lambda )=r} \Theta _\lambda , \end{aligned}$$

(2.26)

is a formal power series in z whose coefficients are finite sums and hence lie in $U_q({\widehat{\mathfrak {n}}}^+) \otimes U_q({\widehat{\mathfrak {n}}}^-)$. Hence $(\Sigma _z \otimes \textsf{id})(\Theta )=(\textsf{id}\otimes \Sigma _{z^{-1}})(\Theta )$ has a well-defined action as a linear-operator-valued formal power series on a tensor product of any $U_q({\widehat{\mathfrak {n}}}^+)$-representation with any $U_q({\widehat{\mathfrak {n}}}^-)$-representation. Consider now the grading-shifted universal R-matrix:

$$\begin{aligned} {\mathcal {R}}(z):= (\Sigma _z \otimes \textsf{id})({\mathcal {R}}) = (\textsf{id}\otimes \Sigma _{z^{-1}})({\mathcal {R}}). \end{aligned}$$

(2.27)

Note that by applying $\Sigma _z \otimes \textsf{id}$ to (2.15) we deduce that ${\mathcal {R}}(z)$ commutes with $\Delta (k_1) = \Delta ^\textsf{op}(k_1) = k_1 \otimes k_1$. We collect the results obtained thus far, writing

$$\begin{aligned} M[[z]] = M \otimes \mathbb {C}[[z]] \end{aligned}$$

for any complex vector space M (in particular, any complex unital associative algebra).

Theorem 2.4

Consider a pair of level-0 representations $\pi ^\pm : U_q({\widehat{\mathfrak {b}}}^\pm ) \rightarrow {{\,\textrm{End}\,}}(V^\pm )$. Then³

$$\begin{aligned} {\mathcal {R}}_{\pi ^+\pi ^-}(z):= (\pi ^+ \otimes \pi ^-)({\mathcal {R}}(z)) \in {{\,\textrm{End}\,}}(V^+ \otimes V^-)[[z]] \end{aligned}$$

(2.28)

is well-defined and commutes with $(\pi ^+ \otimes \pi ^-)(\Delta (k_1)) = \pi ^+(k_1) \otimes \pi ^-(k_1)$.

From now on we will use the standard convention that if $\pi$ is any level-0 representation then the corresponding grading-shifted representation is denoted by a subscript z:

$$\begin{aligned} \pi _z:= \pi \circ \Sigma _z. \end{aligned}$$

(2.29)

Hence we may write

$$\begin{aligned} {\mathcal {R}}_{\pi ^+\pi ^-}(z) = (\pi ^+_z \otimes \pi ^-)({\mathcal {R}}) = (\pi ^+ \otimes \pi ^-_{1/z})({\mathcal {R}}). \end{aligned}$$

Consider two indeterminates $z_1,z_2$. Applying, say, $\Sigma _{z_1} \otimes \textsf{id}\otimes \Sigma _{1/z_2}$, to (2.17), we obtain a $\mathbb {C}[[z_1,z_2]]$-version of the universal Yang–Baxter equation which can be evaluated on suitable triple tensor products.

Proposition 2.5

If $\pi ^+: U_q({\widehat{\mathfrak {b}}}^+) \rightarrow {{\,\textrm{End}\,}}(V^+)$, $\pi : U_q({\widehat{\mathfrak {g}}}) \rightarrow {{\,\textrm{End}\,}}(V)$ and $\pi ^-: U_q({\widehat{\mathfrak {b}}}^-) \rightarrow {{\,\textrm{End}\,}}(V^-)$ are level-0 representations, then we have the following identity of linear-operator-valued formal power series in two indeterminates:

$$\begin{aligned} {\mathcal {R}}_{\pi ^+\pi }(z_1)_{12} \; {\mathcal {R}}_{\pi ^+\pi ^-}(z_1z_2)_{13} \; {\mathcal {R}}_{\pi \pi ^-}(z_2)_{23} \!=\! {\mathcal {R}}_{\pi \pi ^-}(z_2)_{23} \; {\mathcal {R}}_{\pi ^+\pi ^-}(z_1z_2)_{13} \; {\mathcal {R}}_{\pi ^+\pi }(z_1)_{12}.\nonumber \\ \end{aligned}$$

(2.30)

Given a pair of level-0 representations $\pi ^\pm : U_q({\widehat{\mathfrak {b}}}^\pm ) \rightarrow {{\,\textrm{End}\,}}(V^\pm )$ it is often convenient to have an explicit expression of ${\mathcal {R}}_{\pi ^+\pi ^-}(z)$ which does not rely on computing the coefficients of the series ${\mathcal {R}}(z)$. Essentially following Jimbo’s approach from [Ji86b], we may try to solve a linear equation for ${\mathcal {R}}_{\pi ^+\pi ^-}(z)$. To derive such a linear equation, it is convenient to assume that, say, $\pi ^+$ extends to a representation of $U_q({\widehat{\mathfrak {g}}})$. In this case⁴, one directly obtains the following result.

Proposition 2.6

If $\pi ^+$ is a level-0 $U_q({{\widehat{\mathfrak {g}}}})$-representation and $\pi ^-$ a level-0 $U_q({{\widehat{\mathfrak {b}}}}^-)$-representation, then for all $u \in U_q({{\widehat{\mathfrak {b}}}}^-)$ we have

$$\begin{aligned} {\mathcal {R}}_{\pi ^+\pi ^-}(z) \cdot (\pi ^+_z \otimes \pi ^-)(\Delta (u)) = (\pi ^+_z \otimes \pi ^-)(\Delta ^{\textsf {op}}(u)) \cdot {\mathcal {R}}_{\pi ^+\pi ^-}(z). \end{aligned}$$

(2.31)

Obviously there is a counterpart of Proposition 2.6 with the role of $U_q({{\widehat{\mathfrak {b}}}}^-)$ replaced by $U_q({{\widehat{\mathfrak {b}}}}^+)$.

Remark 2.7

If the solution space of the linear equation (2.31) is 1-dimensional, Proposition 2.6 implies that any solution must be a scalar multiple of ${\mathcal {R}}_{\pi ^+\pi ^-}(z)$ and hence satisfy the Yang–Baxter equation. This is well-known if both $V^\pm$ extend to finite-dimensional $U_q({{\widehat{\mathfrak {g}}}})$-modules. In this case the existence of the universal R-matrix implies the existence of a solution of the intertwining condition (2.31) depending rationally on z. If $\pi ^+$ and $\pi ^-$ are also both irreducible then it is known, see e.g. [KS95, Sec. 4.2] and [Ch02, Thm. 3], that $V^+((z)) \otimes V^-$ is irreducible as a representation of $U_q({{\widehat{\mathfrak {g}}}})((z))$ (extension of scalars to formal Laurent series); hence an application of Schur’s lemma yields the 1-dimensionality of the solution space of (2.31). In this case, the rational intertwiner is called trigonometric R-matrix. For more background and detail, see e.g. [He19] and [AV22b, Secs. 2.6 & 2.7].

In the absence of a linear relation such as (2.31), one can use the Yang–Baxter equation (2.30) to determine an explicit expression for one of ${\mathcal {R}}_{\pi ^+\pi }(z)$, ${\mathcal {R}}_{\pi ^+\pi ^-}(z)$, or ${\mathcal {R}}_{\pi \pi ^-}(z)$, provided the other two are known.

2.9 Adjusting the grading

In this approach the use of the principal grading in Theorem 2.4 avoids further constraints on the representations (e.g. local finiteness conditions). For completeness we briefly explain how to extend the results of Sect. 2.8 to arbitrary grading. For nonnegative integers $s_0,s_1$ such that $s_0+s_1$ is nonzero, define a more general Hopf algebra automorphism $\Sigma ^{s_0,s_1}_z$ of $U_q({\widetilde{\mathfrak {g}}})[z,z^{-1}]$ by

$$\begin{aligned} \Sigma ^{s_0,s_1}_z(e_i) = z^{s_i} e_i, \qquad \Sigma ^{s_0,s_1}_z(f_i) = z^{-s_i} f_i, \qquad \Sigma ^{s_0,s_1}_z|_{U_q({\widetilde{\mathfrak {h}}})} = \textsf{id}\end{aligned}$$

(2.32)

(note that the choice $s_0=0$, $s_1=1$ is used in in [KT14, Eq. (2.11)]).

Rather than giving generalized versions of the main results above and of various statements in the remainder of this work, we make an observation which will allow the reader to generate these statements, as required. Recalling the decomposition (2.22) and the associated terminology, suppose the level-0 $U_q({{\widehat{\mathfrak {b}}}}^+)$-module V is generated by a nonzero element of $V(c_0)$ for some $c_0 \in \mathbb {C}^\times$ (which includes all modules considered in this paper and all irreducible finite-dimensional $U_q({{\widehat{\mathfrak {g}}}})$-modules). Then the support of V, see Sect. 2.7, is contained in $q^{2\mathbb {Z}} c_0$. Now for any indeterminate y and any integer m, let $y^{mD}$ denote the linear map on V which acts on $V(q^{-2m}c_0)[y,y^{-1}]$ as scalar multiplication by $y^m$.

Writing the corresponding representation as $\pi : U_q({{\widehat{\mathfrak {b}}}}^+) \rightarrow {{\,\textrm{End}\,}}(V)$, the more general grading-shifted representation $\pi ^{s_0,s_1}_z:= \pi \circ \Sigma ^{s_0,s_1}_z$ can be related to the representation shifted by the principal grading as follows. Adjoining to the ring $\mathbb {C}[z,z^{-1}]$ a square root Z of z, we have

$$\begin{aligned} \pi ^{s_0,s_1}_z = {{\text {Ad}}}\big ( Z^{(s_0-s_1)D} \big ) \circ \pi _{Z^{s_0+s_1}}, \end{aligned}$$

(2.33)

where on the right-hand side ${{\,\textrm{Ad}\,}}$ stands for ‘conjugation by’. See [AV22b, Sec. 5.2] for essentially the same point in the context of irreducible finite-dimensional $U_q({{\widehat{\mathfrak {g}}}})$-representations.

3 The Augmented q-Onsager Algebra, its Twist and its Universal K-Matrix

In parallel with the previous section, we consider a particular subalgebra of $U_q({{\widehat{\mathfrak {g}}}})$ and extend some recent results on universal K-matrices [AV22a, AV22b] in the context of (possibly infinite-dimensional) level-0 representations of Borel subalgebras of quantum affine $\mathfrak {s}\mathfrak {l}_2$. For a related approach tailored to evaluation representations involving essentially the same subalgebra, see [BT18].

3.1 The twist map $\psi$

We consider the following algebra automorphism and coalgebra antiautomorphism of $U_q({\widetilde{\mathfrak {g}}})$:

$$\begin{aligned} \psi := \omega \circ \Phi . \end{aligned}$$

(3.1)

From (2.18–2.19) and (2.24–2.25), respectively, we immediately deduce

$$\begin{aligned} (\psi \otimes \psi )({\mathcal {R}})&= {\mathcal {R}}_{21}, \end{aligned}$$

(3.2)

$$\begin{aligned} \psi \circ \Sigma _z&= \Sigma _{z^{-1}} \circ \psi . \end{aligned}$$

(3.3)

By the following result, P-symmetric R-matrices (${\mathcal {R}}(z)_{21} = {\mathcal {R}}(z)$) naturally arise in tensor products of representations of the upper and lower Borel subalgebras on the same vector space, provided they are related through $\psi$ and the principal grading is used in the definition of grading-shifted universal R-matrix ${\mathcal {R}}(z)$, see (2.27).

Lemma 3.1

Consider two pairs of level-0 representations $\pi ^\pm , \varrho ^\pm : U_q({{\widehat{\mathfrak {b}}}}^\pm ) \rightarrow {{\,\textrm{End}\,}}(V)$ such that

$$\begin{aligned} \varrho ^\mp = \pi ^\pm \circ \psi . \end{aligned}$$

(3.4)

Then ${\mathcal {R}}_{\pi ^+\pi ^-}(z)_{21} = {\mathcal {R}}_{\varrho ^+\varrho ^-}(z)$.

Proof

Unpacking the definitions (2.28) and (2.27), we have

$$\begin{aligned} {\mathcal {R}}_{\pi ^+\pi ^-}(z)_{21} = \Big ( \big ( (\pi ^+ \otimes \pi ^-) \circ (\Sigma _z \otimes {\textsf {id}}) \big )({\mathcal {R}}) \Big )_{21} = \big ( (\pi ^- \otimes \pi ^+) \circ ({\textsf {id}}\otimes \Sigma _z) \big )\big ({\mathcal {R}}_{21}\big ). \end{aligned}$$

Now using (3.2–3.3) we deduce

$$\begin{aligned} {\mathcal {R}}_{\pi ^+\pi ^-}(z)_{21} = \big ( (\pi ^- \otimes \pi ^+) \circ (\psi \otimes \psi ) \circ ({\textsf {id}}\otimes \Sigma _{z^{-1}}) \big )({\mathcal {R}}). \end{aligned}$$

Applying (3.4) and using (2.28) and (2.27) once again, we obtain ${\mathcal {R}}_{\varrho ^+\varrho ^-}(z)$ as required. $\square$

3.2 The augmented q-Onsager algebra

The map $\psi$ plays an important role in the theory of diagonal matrix solutions with a free parameter of the reflection equation in $U_q({{\widehat{\mathfrak {g}}}})$-modules. Namely, fix a parameter $\xi \in \mathbb {C}^\times$ and consider the following subalgebra of $U_q({{\widehat{\mathfrak {g}}}})$, also called the (embedded) augmented q-Onsager algebra:

$$\begin{aligned} U_q(\mathfrak {k}):= \mathbb {C}\big \langle e_0 - q^{-1} \xi ^{-1} k_0 f_1, \, e_1 - q^{-1} \xi k_1 f_0, \, k_0 k_1^{-1}, \, k_0^{-1} k_1 \big \rangle . \end{aligned}$$

(3.5)

This is a left coideal:

$$\begin{aligned} \Delta (U_q(\mathfrak {k})) \subseteq U_q({{\widehat{\mathfrak {g}}}}) \otimes U_q(\mathfrak {k}). \end{aligned}$$

(3.6)

The automorphism $\psi$ is the trivial q-deformation of a Lie algebra automorphism of ${{\widehat{\mathfrak {g}}}}$, also denoted $\psi$, and $U_q(\mathfrak {k})$ is the ($\xi$-dependent) coideal q-deformation of the universal enveloping algebra of the fixed-point subalgebra $\mathfrak {k}= {{\widehat{\mathfrak {g}}}}^\psi$, in the style of [Ko14] but with opposite conventions.

Remark 3.2

See [VW20, Rmk. 2.3] for more background on this subalgebra. Note that the definition of $U_q(\mathfrak {k})$ in loc. cit. has a misprint: $\xi$ should be replaced by $\xi ^{-1}$.

To connect with the universal K-matrix formalism of [AV22a, AV22b], let ${\widetilde{S}}$ be the bialgebra isomorphism⁵ from $U_q({\widetilde{\mathfrak {g}}})$ to $U_q({\widetilde{\mathfrak {g}}})^\textsf{op,cop}$ (also known as the unitary antipode) defined by the assignments

$$\begin{aligned} {\widetilde{S}}(e_i) = -q k_i^{-1} e_i, \quad {\widetilde{S}}(f_i) = -q^{-1} f_i k_i, \quad {\widetilde{S}}(k_i^{\pm 1}) = k_i^{\mp 1}, \quad {\widetilde{S}}(g^{\pm 1}) = g^{\mp 1}. \end{aligned}$$

(3.7)

Note that ${\widetilde{S}}^2=\textsf{id}$. Now consider⁶ the right coideal subalgebra

$$\begin{aligned} U_q(\mathfrak {k})' = {\widetilde{S}}(U_q(\mathfrak {k})) = \mathbb {C}\langle f_0 - q \xi ^{-1} e_1 k_0^{-1}, f_1 - q \xi e_0 k_1^{-1}, k_0 k_1^{-1}, k_0^{-1} k_1 \rangle \end{aligned}$$

which is the subalgebra considered in [AV22a, Sec. 9.7], forming part of a more general family of right coideal subalgebras (quantum symmetric pair subalgebras) of quantum affine algebras as considered in [Ko14, AV22a, AV22b].

3.3 Universal K-matrix

By [AV22a, Thm. 8.5], $U_q({\widetilde{\mathfrak {g}}})$ is endowed with a so-called standard universal K-matrix, which is an invertible element in a completion of $U_q({\widetilde{\mathfrak {b}}}^+)$ satisfying a twisted $U_q(\mathfrak {k})$-intertwining property and a twisted coproduct formula involving the universal R-matrix⁷

$$\begin{aligned} {\mathcal {R}}' = {\mathcal {R}}_{21}^{-1}. \end{aligned}$$

(3.8)

There is an action of invertible elements of a completion of $U_q({\widetilde{\mathfrak {g}}})$, gauge-transforming the universal K-matrix and the twisting operator simultaneously, see [AV22b, Sec. 3.6]. For the case under consideration, there exists a gauge transformation (a ‘Cartan correction’, see [AV22a, Sec. 8.8]) that brings both the intertwining property and the coproduct formula for the universal K-matrix into a particularly nice form. Moreover, the gauge-transformed universal K-matrix still resides in a completion of $U_q({\widetilde{\mathfrak {b}}}^+)$ and, when shifted by the principal grading, acts as a linear-operator-valued formal power series for all level-0 $U_q({{\widehat{\mathfrak {b}}}}^+)$-modules.

To make this more precise, let $\Omega : {\widetilde{P}} \rightarrow \mathbb {C}^\times$ be any group homomorphism such that $\Omega (\alpha _0)=-\xi$ and $\Omega (\alpha _1)=-\xi ^{-1}$. Now define a function $G': {\widetilde{P}} \rightarrow \mathbb {C}^\times$ by

$$\begin{aligned} G'(\lambda ) = \Omega (\lambda ) q^{-(\Phi (\lambda ),\lambda )/2}. \end{aligned}$$

(3.9)

Note that this is not a group homomorphism. Define the corresponding linear operator acting on $U_q({\widetilde{\mathfrak {h}}})$-weight modules as follows:

$$\begin{aligned} G' \cdot v = G'(\lambda ) v, \qquad v \in V_\lambda , \qquad \lambda \in {\widetilde{P}}. \end{aligned}$$

(3.10)

Analogously to our definition of the factor $\kappa$ of the universal R-matrix, we thus obtain an invertible element $G'$ of the weight completion of $U_q({\widetilde{\mathfrak {g}}})$. Finally, let $\delta =\alpha _0+\alpha _1$ be the basic imaginary root of ${{\widehat{\mathfrak {g}}}}$. Then the following result is a special case of [AV22a, Sec. 9.7], with the coproduct formula a direct consequence of [AV22a, (8.21)].

Proposition 3.3

There exists an invertible element

$$\begin{aligned} \Upsilon ' = \sum _{\lambda \in \mathbb {Z}_{\geqslant 0} \delta } \Upsilon '_\lambda , \qquad \Upsilon '_\lambda \in U_q({\widehat{\mathfrak {n}}}^+)_\lambda , \end{aligned}$$

(3.11)

such that the invertible element

$$\begin{aligned} {\mathcal {K}}':= G' \cdot \Upsilon ' \end{aligned}$$

(3.12)

satisfies

$$\begin{aligned} {\mathcal {K}}' \cdot u&= \psi (u) \cdot {\mathcal {K}}' \qquad \text {for all } u \in U_q(\mathfrak {k})', \end{aligned}$$

(3.13)

$$\begin{aligned} \Delta ({\mathcal {K}}')&= (1 \otimes {\mathcal {K}}') \cdot (\psi \otimes \textsf{id})({\mathcal {R}}') \cdot ({\mathcal {K}}' \otimes 1). \end{aligned}$$

(3.14)

Remark 3.4

In general, a universal K-matrix ${\mathcal {K}}'$ satisfying the simple 3-factor coproduct formula (3.14) is called semi-standard, see [AV22a, Sec. 8.10] and cf. [AV22b, Ex. 3.6.3 (2)]. It corresponds to a particular choice of a twist pair $(\psi ,J)$ where $\psi$ is a bialgebra isomorphism from $U_q({\widetilde{\mathfrak {g}}})$ to $U_q({\widetilde{\mathfrak {g}}})^\textsf{cop}$ (essentially the composition of $\omega$ with a diagram automorphism determined by the coideal subalgebra) and J is the trivial Drinfeld twist $1 \otimes 1$, see [AV22a, Sec. 2.4 and 2.5]. The semi-standard K-matrix is always available; what is rather special in the case of the augmented q-Onsager algebra is that the semi-standard K-matrix coincides with the standard K-matrix (and hence lies in a completion of $U_q({\widetilde{\mathfrak {b}}}^+)$).

Now we transform this formalism [AV22a] for the right coideal subalgebra $U_q(\mathfrak {k})'$, expressed in terms of the universal R-matrix ${\mathcal {R}}'$, to a formalism for the left coideal subalgebra $U_q(\mathfrak {k}) = {\widetilde{S}}(U_q(\mathfrak {k})')$, expressed in terms of the universal R-matrix ${\mathcal {R}}$ as used in this paper. To do this, note that, when going from a $U_q({\widetilde{\mathfrak {g}}})$-weight module to its dual, weights transform as $\lambda \mapsto -\lambda$. This defines the extension of S and ${\widetilde{S}}$ to a map on the weight completion of $U_q({\widetilde{\mathfrak {g}}})$. Therefore ${\widetilde{S}}(\Omega ) = \Omega ^{-1}$ but the non-group-like factor of $G'$ is fixed by ${\widetilde{S}}$. We define $G: {\widetilde{P}} \rightarrow \mathbb {C}^\times$ by

$$\begin{aligned} G(\lambda ):= \Omega (\lambda ) q^{(\Phi (\lambda ),\lambda )/2} \end{aligned}$$

(3.15)

so that $G = {\widetilde{S}}(G')^{-1}$. Also, we set

$$\begin{aligned} \Upsilon := {\widetilde{S}}(\Upsilon ')^{-1} = \sum _{\lambda \in \mathbb {Z}_{\geqslant 0} \delta } \Upsilon _\lambda , \qquad \Upsilon _\lambda \in {\widetilde{S}}(U_q({{\widehat{\mathfrak {n}}}}^+)_\lambda ) \subset U_q({{\widehat{\mathfrak {h}}}}) \cdot U_q({{\widehat{\mathfrak {n}}}}^+)_\lambda . \end{aligned}$$

(3.16)

Proposition 3.5

The element

$$\begin{aligned} {\mathcal {K}}:= {\widetilde{S}}({\mathcal {K}}')^{-1} = G \cdot \Upsilon \end{aligned}$$

(3.17)

satisfies

$$\begin{aligned} {\mathcal {K}}\cdot u&= \psi (u) \cdot {\mathcal {K}}\qquad \text {for all } u \in U_q(\mathfrak {k}), \end{aligned}$$

(3.18)

$$\begin{aligned} \Delta ({\mathcal {K}})&= ({\mathcal {K}}\otimes 1) \cdot (\textsf{id}\otimes \psi )({\mathcal {R}}) \cdot (1 \otimes {\mathcal {K}}). \end{aligned}$$

(3.19)

Proof

This follows straightforwardly from Proposition 3.3. Namely, we apply ${\widetilde{S}}$ to (3.13) and $({\widetilde{S}} \otimes {\widetilde{S}}) \circ \sigma$ to (3.14), and use the fact that ${\widetilde{S}} \circ \psi = \psi \circ {\widetilde{S}}$ and $({\widetilde{S}} \otimes {\widetilde{S}})({\mathcal {R}}) = {\mathcal {R}}$. $\square$

Note that $U_q({{\widehat{\mathfrak {b}}}}^+)$ is a bialgebra and, as expected, the right-hand side of (3.19) lies in a completion of $U_q({{\widehat{\mathfrak {b}}}}^+) \otimes U_q({{\widehat{\mathfrak {b}}}}^+)$, since $\psi$ interchanges the two Borel subalgebras $U_q({{\widehat{\mathfrak {b}}}}^\pm )$. The reflection equation satisfied by the universal element ${\mathcal {K}}$ is as follows:

$$\begin{aligned} {\mathcal {R}}\!\cdot \! ({\mathcal {K}}\otimes 1) \cdot (\textsf{id}\otimes \psi )({\mathcal {R}}) \cdot (1 \otimes {\mathcal {K}}) \!=\! (1 \otimes {\mathcal {K}}) \cdot (\textsf{id}\otimes \psi )({\mathcal {R}}) \cdot ({\mathcal {K}}\otimes 1) \cdot {\mathcal {R}}.\qquad \end{aligned}$$

(3.20)

This is a consequence of the linear relation (2.15) for ${\mathcal {R}}$ and the coproduct formula (3.19) for ${\mathcal {K}}$, alongside (3.2) and $\psi ^2=\textsf{id}$.

3.4 The action of the universal K-matrix on level-0 representations

To deduce that ${\mathcal {K}}$ has a well-defined action on level-0 representations of, say, $U_q({{\widehat{\mathfrak {b}}}}^+)$, we can proceed in a similar way to the case of the R-matrix. This builds on the finite-dimensional theory for more general quantum symmetric pair subalgebras in [AV22b, Sec. 4].

First note that if $\pi$ is a level-0 representation, $\pi$ and the twisted representation $\pi \circ \psi$ coincide on $U_q({{\widehat{\mathfrak {h}}}})$. Now let z once again be a formal variable. Note that by (3.15) the function G sends the basic imaginary root $\delta$ to 1. Hence the proof of [AV22b, Prop. 4.3.1 (3)] implies that the corresponding factor G of the universal K-matrix descends to level-0 modules. Furthermore, the argument that shows $\Sigma _z(\Theta )$ is a $U_q({{\widehat{\mathfrak {n}}}}^+) \otimes U_q({{\widehat{\mathfrak {n}}}}^-)$-valued formal power series can be easily adapted to $\Upsilon$; it yields a formal power series with coefficients in ${\widetilde{S}}(U_q({{\widehat{\mathfrak {n}}}}^+)) \subset U_q({{\widehat{\mathfrak {b}}}}^+)$:

$$\begin{aligned} \Sigma _z(\Upsilon ) = \sum _{r \geqslant 0} z^r \sum _{\lambda \in \mathbb {Z}_{\geqslant 0} \delta , \, {\textsf {ht}}(\lambda )=r} \Upsilon _\lambda . \end{aligned}$$

Now consider the grading-shifted universal K-matrix:

$$\begin{aligned} {\mathcal {K}}(z) = \Sigma _z({\mathcal {K}}). \end{aligned}$$

(3.21)

Noting that the form of $\Upsilon$ implies that ${\mathcal {K}}$ commutes with $k_1$, we arrive at the following main result, which is a boundary analogue of Theorem 2.4.

Theorem 3.6

Consider a level-0 representation $\pi : U_q({{\widehat{\mathfrak {b}}}}^+) \rightarrow {{\,\textrm{End}\,}}(V)$. Then⁸

$$\begin{aligned} {\mathcal {K}}_{\pi }(z):= \pi ({\mathcal {K}}(z)) \in {{\,\textrm{End}\,}}(V)[[z]] \end{aligned}$$

(3.22)

is well-defined and commutes with $\pi (k_1)$.

We will also need boundary counterparts of Propositions 2.5 and 2.6. Consider two indeterminates $z_1,z_2$. Applying $\Sigma _{z_1} \otimes \Sigma _{z_2}$ to (3.20) and using (3.3), we obtain the following reflection equation for the grading-shifted universal operators:

$$\begin{aligned}{} & {} {\mathcal {R}}(z_1/z_2) \cdot ({\mathcal {K}}(z_1) \otimes 1) \cdot (\textsf{id}\otimes \psi )({\mathcal {R}}(z_1z_2)) \cdot (1 \otimes {\mathcal {K}}(z_2)) = \qquad \nonumber \\{} & {} \quad \qquad = (1 \otimes {\mathcal {K}}(z_2)) \cdot (\textsf{id}\otimes \psi )({\mathcal {R}}(z_1z_2)) \cdot ({\mathcal {K}}(z_1) \otimes 1) \cdot {\mathcal {R}}(z_1/z_2). \end{aligned}$$

(3.23)

Recalling that the universal R-matrix ${\mathcal {R}}$ lies in a completion of $U_q({{\widehat{\mathfrak {b}}}}^+) \otimes U_q({{\widehat{\mathfrak {b}}}}^-)$ and applying a tensor product of suitable representations to (3.23), one obtains the right reflection equation with multiplicative spectral parameters for P-symmetric R-matrices, as we now state precisely.

Proposition 3.7

Consider level-0 representations $\pi ^+: U_q({{\widehat{\mathfrak {b}}}}^+) \rightarrow {{\,\textrm{End}\,}}(V^+)$ and $\pi : U_q({{\widehat{\mathfrak {g}}}}) \rightarrow {{\,\textrm{End}\,}}(V)$ such that $\pi \circ \psi = \pi$. Then

$$\begin{aligned}{} & {} {\mathcal {R}}_{\pi ^+\pi }(z_1/z_2) ({\mathcal {K}}_{\pi ^+}(z_1) \otimes \textsf{Id}_V) {\mathcal {R}}_{\pi ^+\pi }(z_1z_2) (\textsf{Id}_{V^+} \otimes {\mathcal {K}}_\pi (z_2)) = \qquad \nonumber \\{} & {} \quad \qquad = (\textsf{Id}_{V^+} \otimes {\mathcal {K}}_\pi (z_2)) {\mathcal {R}}_{\pi ^+\pi }(z_1z_2) ({\mathcal {K}}_{\pi ^+}(z_1) \otimes \textsf{Id}_V) {\mathcal {R}}_{\pi ^+\pi }(z_1/z_2). \end{aligned}$$

(3.24)

The use of linear relations to find explicit solutions of reflection equations was proposed in [MN98, DG02, DM03]. As before, we assume that $\pi$ extends to a $U_q({{\widehat{\mathfrak {g}}}})$-representation,⁹ in which case it restricts to a $U_q(\mathfrak {k})$-representation and we obtain the following result as a consequence of (3.3).

Proposition 3.8

If $\pi : U_q({{\widehat{\mathfrak {g}}}}) \rightarrow {{\,\textrm{End}\,}}(V)$ is a level-0 representation such that $\pi \circ \psi = \pi$, then

$$\begin{aligned} {\mathcal {K}}_\pi (z) \cdot \pi _z(u) = \pi _{1/z}(u) \cdot {\mathcal {K}}_\pi (z) \qquad \text {for all } u \in U_q(\mathfrak {k}). \end{aligned}$$

(3.25)

We close this section with some comments parallel to Remark 2.7.

Remark 3.9

If the solution space of (3.25) is 1-dimensional, Proposition 3.8 implies that any solution must be a scalar multiple of $\mathcal K(z)$ and hence automatically satisfy the reflection equation (3.24). In the case that $\pi : U_q({{\widehat{\mathfrak {b}}}}^+) \rightarrow {{\,\textrm{End}\,}}(V)$ extends to a representation and V is finite-dimensional, there is an analogue to Remark 2.7. Namely, the solution space of (3.25) for irreducible representations is 1-dimensional and the existence of a solution of the intertwining condition (3.25) depending rationally on z leads to a trigonometric K-matrix. See [AV22b, Secs. 5 and 6] for more detail.

To explicitly determine ${\mathcal {K}}_{\pi ^+}(z)$ in the cases where $\pi ^+: U_q({{\widehat{\mathfrak {b}}}}^+) \rightarrow {{\,\textrm{End}\,}}(V)$ does not extend to a $U_q({{\widehat{\mathfrak {g}}}})$-representation, we will use the reflection equation (3.24), with the other K-matrix ${\mathcal {K}}_\pi (z)$ determined using Proposition 3.8.

4 Borel Representations in Terms of the q-Oscillator Algebra

4.1 The infinite-dimensional vector space W

The countably-infinite-dimensional vector space plays a central role in the theory of Baxter’s Q-operators. We may define it as the free $\mathbb {C}$-module over a given set $\{ w_j \}_{j \in \mathbb {Z}_{\geqslant 0}}$:

$$\begin{aligned} W = \bigoplus _{j \geqslant 0} \mathbb {C}w_j. \end{aligned}$$

(4.1)

Given this distinguished basis, elements of ${{\,\textrm{End}\,}}(W)$ naturally identify with infinite-by-infinite matrices with the property that all but finitely many entries of each column are zero.

It is convenient to work with a particular subalgebra of ${{\,\textrm{End}\,}}(W)$ depending on the deformation parameter q. More precisely, consider the $\mathbb {C}$-linear maps a, $a^\dag$ on W defined by

$$\begin{aligned} a \cdot w_{j+1} = w_j, \qquad a \cdot w_0 = 0, \qquad a^\dag \cdot w_j = \big ( 1-q^{2(j+1)} \big ) w_{j+1} \end{aligned}$$

(4.2)

for all $j \in \mathbb {Z}_{\geqslant 0}$. For the description of L-operators associated to $U_q({{\widehat{\mathfrak {g}}}})$ acting on $W \otimes \mathbb {C}^2$ (particular solutions of the Yang–Baxter equation), it is convenient to consider a linear operator $q^D$ which is a square root of $1-a^\dag a$, i.e. $q^D \cdot w_j = q^j w_j$ for $j \in \mathbb {Z}_{\geqslant 0}$. Note that $q^D$ is invertible and we let $q^{-D}$ denote its inverse.

Remark 4.1

Often the q-oscillator algebra is defined as an abstract algebra, generated by a, $a^\dag$ and $q^{\pm D}$ subject to certain relations, which naturally embeds into ${{\,\textrm{End}\,}}(W)$. This version of the q-oscillator algebra appeared in the guise of a topological algebra for instance in [BGKNR10, Sec. 2.3] and with slightly different conventions in [KT14]¹⁰.

4.2 Diagonal operators from functions and an extended q-oscillator algebra

To accommodate the action of the universal R and K-matrices on certain level-0 modules, we will need an extension of the commutative subalgebra $\langle q^{\pm D} \rangle$ and work over the commutative ring $\mathbb {C}[[z]]$.

Denote by ${\mathcal {F}}$ the commutative algebra of functions from $\mathbb {Z}_{\geqslant 0}$ to $\mathbb {C}[[z]]$. For any $f \in {\mathcal {F}}$ we define $f(D) \in {{\,\textrm{End}\,}}(W)[[z]]$ via

$$\begin{aligned} f(D) \cdot w_j = f(j) w_j. \end{aligned}$$

(4.3)

Thus, we obtain an algebra embedding ${\mathcal {F}}\rightarrow {{\,\textrm{End}\,}}(W)[[z]]$. Now we combine this with the maps a, $a^\dag$ defined above (viewed as maps on $W \otimes \mathbb {C}[[z]]$, acting trivially on the second factor).

Definition 4.2

The (extended) q-oscillator algebra is the subalgebra ${\mathcal {A}}\subset {{\,\textrm{End}\,}}(W)[[z]]$ generated by $a^\dag$, a and ${\mathcal {F}}(D)$.

As can be verified on basis vectors, in ${\mathcal {A}}$ one has the relations

$$\begin{aligned} a a^\dag&= {} 1-q^{2(D+1)},&\qquad a^\dag a&= 1-q^{2D}, \nonumber \\ a f(D)&= f(D+1) a,&\qquad f(D) a^\dag&= {} a^\dag f(D+1). \end{aligned}$$

(4.4)

One straightforwardly verifies that the subalgebras ${\mathcal {F}}(D)$, $\langle a^\dag \rangle$ and $\langle a \rangle$ are self-centralizing. Note that the operator

$$\begin{aligned} {{\bar{a}}}^\dag := -q^{-2D} a^\dag \in {{\,\textrm{End}\,}}(W) \end{aligned}$$

(4.5)

sends $w_j$ to $(1-q^{-2(j+1)})w_{j+1}$. Clearly, ${\mathcal {A}}$ is also generated by ${{\bar{a}}}^\dag$, a and ${\mathcal {F}}(D)$. The transformation $q \mapsto q^{-1}$ defines an algebra automorphism of ${\mathcal {A}}$, preserving the subalgebra ${\mathcal {F}}(D)$, fixing the generator a and interchanging the generators $a^\dag$ and ${{\bar{a}}}^\dag$.

4.3 Endomorphisms of $W \otimes W$

The linear maps

$$\begin{aligned} a_1:=a \otimes \textsf{Id}_W, \qquad a^\dag _1:= a^\dag \otimes \textsf{Id}_W, \qquad a_2:= \textsf{Id}_W \otimes a, \qquad a^\dag _2:= \textsf{Id}_W \otimes a^\dag \end{aligned}$$

together with ${\mathcal {F}}(D_1) \cup {\mathcal {F}}(D_2)$ generate ${\mathcal {A}}\otimes {\mathcal {A}}$ over $\mathbb {C}[[z]]$. We will need a larger subalgebra of ${{\,\textrm{End}\,}}(W \otimes W)$: we will allow all functions of two nonnegative integers as well as formal power series in certain locally nilpotent endomorphisms.

Denote by ${\mathcal {F}}^{(2)}$ the commutative algebra of functions from $\mathbb {Z}_{\geqslant 0} \times \mathbb {Z}_{\geqslant 0}$ to $\mathbb {C}[[z]]$. For any $f \in {\mathcal {F}}^{(2)}$ we define $f(D_1,D_2) \in {{\,\textrm{End}\,}}(W \otimes W)[[z]]$ via

$$\begin{aligned} f(D_1,D_2) \cdot (w_j \otimes w_k) = f(j,k) w_j \otimes w_k, \end{aligned}$$

(4.6)

yielding an algebra embedding ${\mathcal {F}}^{(2)} \rightarrow {{\,\textrm{End}\,}}(W \otimes W)[[z]]$. Now note that $a_1 a^\dag _2$ and $a^\dag _1 a_2$ are locally nilpotent endomorphisms of $W \otimes W$. Hence, for any $g_{k,\ell }, h_{k,\ell } \in {\mathcal {F}}^{(2)}$ series of the form

$$\begin{aligned} \sum _{k,\ell \geqslant 0} (a^\dag _2)^\ell g_{k,\ell }(D_1,D_2) a_1^k, \qquad \sum _{k,\ell \geqslant 0} (a^\dag _1)^k h_{k,\ell }(D_1,D_2) a_2^\ell \end{aligned}$$

(4.7)

truncate when applied to any basis vector $w_j \otimes w_{j'}$. We obtain a class of well-defined elements of ${{\,\textrm{End}\,}}(W \otimes W)[[z]]$. We denote by ${\mathcal {A}}^{(2)}$ the $\mathbb {C}[[z]]$-span of the operator-valued formal series (4.7), which is easily seen to be a subalgebra of ${{\,\textrm{End}\,}}(W \otimes W)[[z]]$.

4.4 The Borel representations

We introduce four level-0 representations of $U_q({{\widehat{\mathfrak {b}}}}^+)$. First of all, let $\mu \in \mathbb {C}$ be a free parameter. It is straightforward to check that the following assignments define a representation $\upsilon$ of $U_q({{\widehat{\mathfrak {g}}}})$ on W:

$$\begin{aligned} \upsilon (e_0)= & {} \upsilon (f_1) = \frac{1}{1-q^2} a^\dag , \qquad \qquad \qquad \qquad \qquad \upsilon (k_0) = q^{-\mu +1+2D}, \nonumber \\ \upsilon (e_1)= & {} \upsilon (f_0) = \frac{q^2}{1-q^2} a (q^{-\mu } - q^{\mu -2D}), \qquad \upsilon (k_1) \,= q^{\mu -1-2D}. \end{aligned}$$

(4.8)

The module structure on W defined by $\upsilon$ is the evaluation Verma module: affinizations of finite-dimensional irreducible $U_q(\mathfrak {s}\mathfrak {l}_2)$-modules arise as quotients if $\mu \in \mathbb {Z}_{>0}$ (also see [KT14, Sec. 2.2]).

We will in addition consider three $U_q({\widehat{\mathfrak {b}}}^+)$-representations which do not extend to representations of $U_q({\widehat{\mathfrak {g}}})$. A useful reducible representation $\phi : U_q({\widehat{\mathfrak {b}}}^+) \rightarrow {{\,\textrm{End}\,}}(W)$ is given by

$$\begin{aligned} \phi (e_0) = 0, \quad \phi (e_1) = \frac{q}{1-q^2} a, \quad \phi (k_0) = q^{\mu +1+2D}, \quad \phi (k_1) = q^{-\mu -1-2D}\nonumber \\ \end{aligned}$$

(4.9)

which is closely related to the special evaluation homomorphism defined in [KT14, Eq. (4.6)]. The following representations $\varrho , {{\bar{\varrho }}}: U_q({\widehat{\mathfrak {b}}}^+) \rightarrow {{\,\textrm{End}\,}}(W)$ play an essential role in the definition of Baxter Q-operators:

$$\begin{aligned} \varrho (e_0)&= {} \frac{1}{1-q^2} a^\dag , \quad&\varrho (e_1)&= \frac{q^2}{1-q^2} a, \quad&\varrho (k_0)&= q^{2D}, \quad&\varrho (k_1)&= q^{-2D}, \nonumber \\ {{\bar{\varrho }}}(e_0)&= {} \frac{q^2}{1-q^2} {{\bar{a}}}^\dag , \quad \quad&{{\bar{\varrho }}}(e_1)&= \frac{1}{1-q^2} a, \quad \quad&{{\bar{\varrho }}}(k_0)&= q^{2(D+1)}, \quad \quad&{{\bar{\varrho }}}(k_1)&= q^{-2(D+1)}.\nonumber \\ \end{aligned}$$

(4.10)

They correspond to the representations $L^\pm _{1,a}$ introduced in [HJ12, Def. 3.7] for suitable $a \in \mathbb {C}^\times$ (called prefundamental representations in [FH15] which considers their role in the construction of Q-operators for closed chains).

We will henceforth repeatedly denote grading-shifted representations by the notation (2.29). Note that the grading-shifted representation $\upsilon _z$ is an algebra homomorphism from $U_q({{\widehat{\mathfrak {g}}}})$ to ${{\,\textrm{End}\,}}(W)[z,z^{-1}]$. Furthermore, the grading-shifted representations $\upsilon _z|_{U_q({\widehat{\mathfrak {b}}}^+)}$, $\phi _z$, $\varrho _z$, ${{\bar{\varrho }}}_z$ are algebra homomorphisms from $U_q({\widehat{\mathfrak {b}}}^+)$ to ${{\,\textrm{End}\,}}(W)[z] \subset {{\,\textrm{End}\,}}(W)[[z]]$. Finally, note that $\varrho _z$, ${{\bar{\varrho }}}_z$ correspond to the representations defined by [KT14, Eq. (3.5)].

Remark 4.3

The grading-shifted representation in [VW20, Eq. (2.9)] is related to $\varrho _z$ by a factor of $-1$ in the actions of $e_0$ and $e_1$: in other words it is equal to $\varrho _{-z}$. Since the Baxter Q-operators only depend on $z^2$, see [VW20, Lem. 4.5], there are no serious discrepancies. The benefit of the current choice is its consistency across the relevant level-0 representations, with $\upsilon$ having the same sign convention as finite-dimensional representations such as $\Pi$, see Sect. 5.

4.5 The $U_q({{\widehat{\mathfrak {b}}}}^+)$-intertwiner ${\mathcal {O}}$

The tensor products $\varrho _{q^{-\mu /2} z} \otimes {{\bar{\varrho }}}_{q^{\mu /2}z}$ and $\upsilon _{z} \otimes \phi _{z}$ of shifted representations are closely related in the following sense: the two induced $U_q({{\widehat{\mathfrak {b}}}}^+)$-actions on $W \otimes W$ are conjugate by an element in ${\mathcal {A}}^{(2)}$ which is independent of z. More precisely, consider the deformed exponential

$$\begin{aligned} e_{q^2}(x) = \sum _{k=0}^\infty \frac{x^k}{(q^2;q^2)_k}. \end{aligned}$$

(4.11)

We refer to Appendix A for more detail. We now define the following element of ${{\,\textrm{GL}\,}}(W \otimes W)$:

$$\begin{aligned} {\mathcal {O}}= e_{q^2}(q^2 a_1 {{\bar{a}}}^\dag _2)^{-1} q^{\mu (D_1-D_2)/2}. \end{aligned}$$

(4.12)

The following statement is [KT14, Eq. (4.4)] and connects to [FH15, Thm. 3.8]; for completeness we provide a proof in the present conventions.

Theorem 4.4

The $U_q({{\widehat{\mathfrak {b}}}}^+)$-representations $\varrho _{q^{-\mu /2} z} \otimes {{\bar{\varrho }}}_{q^{\mu /2} z}$ and $\upsilon _{z} \otimes \phi _{z}$ are intertwined by ${\mathcal {O}}$:

$$\begin{aligned} {\mathcal {O}}\, \big ( \varrho _{q^{-\mu /2} z} \otimes {{\bar{\varrho }}}_{q^{\mu /2} z} \big )(\Delta (u)) = \big ( \upsilon _{z} \otimes \phi _{z} \big )(\Delta (u)) \,\, {\mathcal {O}}\qquad \text {for all } u \in U_q({{\widehat{\mathfrak {b}}}}^+). \end{aligned}$$

(4.13)

Proof

The relations (A.13–A.15) can be evaluated at $y=q^2$, yielding

$$\begin{aligned}&q^{\mu (D_2-D_1)/2} e_{q^2}(q^2a_1 {{\bar{a}}}^\dag _2) {{\bar{a}}}^\dag _2 = \big ( q^{-\mu /2} a^\dag _1 + q^{2(D_1+1)+\mu /2} {{\bar{a}}}^\dag _2 \big ) q^{\mu (D_2-D_1)/2} e_{q^2}(q^2a_1 {{\bar{a}}}^\dag _2), \\&q^{\mu (D_2-D_1)/2} e_{q^2}(q^2a_1 {{\bar{a}}}^\dag _2) \big ( a_1 (q^{-2\mu } - q^{- 2D_1}) + q^{-2(D_1+1)} a_2 \big )= \qquad \qquad \\&\qquad \qquad \qquad \qquad = \big ( a_1 q^{-3\mu /2} + q^{-\mu /2 - 2(D_1+1)} a_2 \big ) q^{\mu (D_2-D_1)/2} e_{q^2}(q^2a_1 {{\bar{a}}}^\dag _2), \\&q^{\mu (D_2-D_1)/2} e_{q^2}(q^2a_1 {{\bar{a}}}^\dag _2) q^{2(D_1+D_2+1)} = q^{2(D_1+D_2+1)} q^{\mu (D_2-D_1)/2} e_{q^2}(q^2 a_1 {{\bar{a}}}^\dag _2) ,\\&q^{\mu (D_2-D_1)/2} e_{q^2}(q^2 a_1 {{\bar{a}}}^\dag _2) q^{-2(D_1+D_2+1)} = q^{-2(D_1+D_2+1)} q^{\mu (D_2-D_1)/2} e_{q^2}(q^2 a_1 {{\bar{a}}}^\dag _2). \end{aligned}$$

These directly imply (4.13) for $u \in \{ e_0,e_1,k_0,k_1\}$. $\square$

4.6 Formalism for $U_q({{\widehat{\mathfrak {b}}}}^-)$

Recall from (3.1) the automorphism $\psi$ which interchanges the two Borel subalgebras. Note that the representation $\upsilon : U_q({{\widehat{\mathfrak {g}}}}) \rightarrow {{\,\textrm{End}\,}}(W)$ satisfies

$$\begin{aligned} \upsilon = \upsilon \circ \psi . \end{aligned}$$

(4.14)

Hence, it is natural to define representations of $U_q({{\widehat{\mathfrak {b}}}}^-)$ corresponding to $\varrho$, ${{\bar{\varrho }}}$ and $\phi$, as follows:

$$\begin{aligned} \varrho ^-:= \varrho \circ \psi , \qquad {{\bar{\varrho }}}^{\, -}:= {{\bar{\varrho }}}\circ \psi , \qquad \phi ^-:= \phi \circ \psi . \end{aligned}$$

(4.15)

Explicitly, we have

$$\begin{aligned} \varrho ^-(f_0)&= {} \frac{q^2}{1-q^2} a, \quad&\varrho ^-(f_1)&= \frac{1}{1-q^2} a^\dag , \quad&\varrho ^-(k_0)&= q^{2D},&\varrho ^-(k_1)&= q^{-2D}, \nonumber \\ {{\bar{\varrho }}}^{\, -}(f_0)&= {} \frac{1}{1-q^2} a,&{{\bar{\varrho }}}^{\, -}(f_1)&= \frac{q^2}{1-q^2} {{\bar{a}}}^\dag ,&{{\bar{\varrho }}}^{\, -}(k_0)&= q^{2(D+1)},&{{\bar{\varrho }}}^{\, -}(k_1)&= q^{-2(D+1)}, \nonumber \\ \phi ^-(f_0)&= {} \frac{q}{1-q^2} a,&\phi ^-(f_1)&= 0,&\phi ^-(k_0)&= q^{\mu +1+2D},&\quad \phi ^-(k_1)&= q^{-\mu -1-2D}.\nonumber \\ \end{aligned}$$

(4.16)

By (3.3), whereas the grading-shifted representations $\varrho _z$, ${{\bar{\varrho }}}_z$, $\phi _z$ take values in ${{\,\textrm{End}\,}}(W) \otimes \mathbb {C}[z]$, their negative counterparts $\varrho ^-_z$, ${{\bar{\varrho }}}^{\, -}_z$, $\phi ^-_z$ take values in ${{\,\textrm{End}\,}}(W) \otimes \mathbb {C}[z^{-1}]$.

Since $\psi$ is a coalgebra antiautomorphism, using (3.3) we immediately deduce the following characterization of the tensorial opposite of the intertwiner ${\mathcal {O}}$.

Corollary 4.5

The linear map

$$\begin{aligned} {\mathcal {O}}_{21} = e_{q^2}(q^2{{\bar{a}}}^\dag _1 a_2)^{-1} q^{\mu (D_2-D_1)/2} \in {{\,\textrm{End}\,}}(W \otimes W). \end{aligned}$$

(4.17)

intertwines the $U_q({\widehat{\mathfrak {b}}}^-)$-representations ${{\bar{\varrho }}}^{\, -}_{q^{-\mu /2} z} \otimes \varrho ^-_{q^{\mu /2} z}$ and $\phi ^-_z \otimes \upsilon _z$, viz.

$$\begin{aligned} {\mathcal {O}}_{21} \, \big ( {{\bar{\varrho }}}^{\, -}_{q^{-\mu /2} z} \otimes \varrho ^-_{q^{\mu /2} z} \big )(\Delta (u)) = \big ( \phi ^-_z \otimes \upsilon _z \big )(\Delta (u)) \, \, {\mathcal {O}}_{21} \quad \text{ for } \text{ all } u \in U_q({\widehat{\mathfrak {b}}}^-).\nonumber \\ \end{aligned}$$

(4.18)

5 L-Operators and R-Operators

In order to define L-operators, we recall the standard 2-dimensional representation $\Pi : U_q({\widehat{\mathfrak {g}}}) \rightarrow {{\,\textrm{End}\,}}(\mathbb {C}^2)$ determined by

$$\begin{aligned} \Pi (e_0) = \Pi (f_1)= & {} \begin{pmatrix} 0 &{} 0 \\ 1 &{} 0 \end{pmatrix}, \quad \quad \Pi (k_0) = \begin{pmatrix} q^{-1} &{} 0 \\ 0 &{} q \end{pmatrix}, \nonumber \\ \Pi (e_1) = \Pi (f_0)= & {} \begin{pmatrix} 0 &{} 1 \\ 0 &{} 0 \end{pmatrix}, \quad \quad \Pi (k_1) = \begin{pmatrix} q &{} 0 \\ 0 &{} q^{-1} \end{pmatrix}. \end{aligned}$$

(5.1)

In analogy with (4.14), we have

$$\begin{aligned} \Pi = \Pi \circ \psi . \end{aligned}$$

(5.2)

5.1 L-operators for $U_q({{\widehat{\mathfrak {b}}}}^+)$-modules

We will now obtain explicit formulas for certain scalar multiples of the four different actions of the grading-shifted universal R-matrix on $W \otimes \mathbb {C}^2$. In these cases both Theorem 2.4 and Proposition 2.6 apply. It turns out that the relevant linear equations all have 1-dimensional solution spaces over $\mathbb {C}[[z]]$. The following linear operators are convenient scalar multiples.

$$\begin{aligned} {\mathcal {L}}_\varrho (z)&= \begin{pmatrix} q^D &{} a^\dag q^{-D-1} z \\ a q^{D+1} z &{} q^{-D}-q^{D+2} z^2 \end{pmatrix}, \end{aligned}$$

(5.3)

$$\begin{aligned} {\mathcal {L}}_{{\bar{\varrho }}}(z)&= \begin{pmatrix} q^{D+1} - q^{-D+1} z^2 &{} {{\bar{a}}}^\dag q^{-D} z \\ a q^{D} z &{} q^{-D-1} \end{pmatrix}, \end{aligned}$$

(5.4)

$$\begin{aligned} {\mathcal {L}}_\upsilon (z)&= \begin{pmatrix} q^D - q^{-D+\mu } z^2 &{} a^\dag q^{-D-2+\mu } z \\ a q \big ( q^{D-\mu } - q^{-D+\mu } \big ) z &{} q^{-D-1+\mu } - q^{D+1} z^2 \end{pmatrix}, \end{aligned}$$

(5.5)

$$\begin{aligned} {\mathcal {L}}_\phi (z)&= \begin{pmatrix} q^{D+1} &{} 0 \\ a q^{D+1} z &{} q^{-D-\mu } \end{pmatrix}. \end{aligned}$$

(5.6)

Remark 5.1

We have abused notation by representing linear operators on ${{\,\textrm{End}\,}}(W \otimes \mathbb {C}^2)$ as $2 \times 2$ matrices with coefficients in ${{\,\textrm{End}\,}}(W)$ (as opposed to the conventional usage that realizes operators on ${{\,\textrm{End}\,}}(\mathbb {C}^2\otimes W)$ in this way).

The following result is [KT14, Cor. 4.2].

Theorem 5.2

The above L-operators satisfy the following relation in ${{\,\textrm{End}\,}}(W \otimes W \otimes \mathbb {C}^2)[[z]]$:

$$\begin{aligned} {\mathcal {O}}_{12} {\mathcal {L}}_\varrho (q^{-\mu /2} z)_{13} {\mathcal {L}}_{{\bar{\varrho }}}(q^{\mu /2} z)_{23} = {\mathcal {L}}_\upsilon (z)_{13} {\mathcal {L}}_\phi (z)_{23} {\mathcal {O}}_{12}. \end{aligned}$$

(5.7)

Proof

From (2.16) one deduces

$$\begin{aligned} {\mathcal {L}}_\varrho (q^{-\mu /2} z)_{13} {\mathcal {L}}_{{\bar{\varrho }}}(q^{\mu /2} z)_{23} \;&\propto \; (\varrho _{q^{-\mu /2} z} \otimes {{\bar{\varrho }}}_{q^{\mu /2} z} \otimes \Pi )\big ((\Delta \otimes \textsf{id})({\mathcal {R}})\big ), \\ {\mathcal {L}}_\upsilon (z)_{13} {\mathcal {L}}_\phi (z)_{23} \;&\propto \; (\upsilon _{z} \otimes \phi _{z} \otimes \Pi )\big ((\Delta \otimes \textsf{id})({\mathcal {R}})\big ). \end{aligned}$$

Now Theorem 4.4 implies (5.7) up to a scalar. By applying both sides to $w_0 \otimes w_0 \otimes ({1 \atop 0})$ one observes that the scalar is 1. $\square$

Given the L-operators for the various $U_q({{\widehat{\mathfrak {b}}}}^+)$-representations, Lemma 3.1 provides us with L-operators for the corresponding $U_q({{\widehat{\mathfrak {b}}}}^-)$-representations: ${\mathcal {L}}^-_\pi (z) = {\mathcal {L}}_\pi (z)_{21}$ for $\pi \in \{ \varrho , {{\bar{\varrho }}}, \upsilon , \phi \}$. These are scalar multiples of ${\mathcal {R}}_{\Pi \varrho ^-}(z)$, ${\mathcal {R}}_{\Pi {{\bar{\varrho }}}^{\, -}}(z)$, ${\mathcal {R}}_{\Pi \upsilon }(z)$ and ${\mathcal {R}}_{\Pi \phi ^-}(z)$, respectively. Theorem 5.2 immediately yields the following result:

Corollary 5.3

The following relation in ${{\,\textrm{End}\,}}(\mathbb {C}^2 \otimes W \otimes W)[[z]]$ is satisfied:

$$\begin{aligned} {\mathcal {O}}_{32} {\mathcal {L}}^-_{\varrho }(q^{-\mu /2} z)_{13} {\mathcal {L}}^-_{{{\bar{\varrho }}}}(q^{\mu /2} z)_{12} = {\mathcal {L}}^-_{\upsilon }(z)_{13} {\mathcal {L}}^-_{\phi }(z)_{12} {\mathcal {O}}_{32}. \end{aligned}$$

(5.8)

5.2 Actions of ${\mathcal {R}}(z)$ on tensor products of infinite-dimensional Borel representations

By Theorem 2.4, the grading-shifted universal R-matrix has well-defined actions on the tensor product of the level-0 modules $(\upsilon ,W)$ and $(\phi ^-,W)$ and on the tensor product of the level-0 modules $(\varrho ,W)$ and $({{\bar{\varrho }}}^{\, -},W)$ as ${{\,\textrm{End}\,}}(W \otimes W)$-valued formal power series. Note that, using the terminology of Sect. 2.7, $\mathbb {C}w_0 \otimes w_0 \subset W \otimes W$ is the subspace of weight $q^{-2}$ and hence $w_0 \otimes w_0$ is an eigenvector of both actions of the universal R-matrix with invertible eigenvalues. It is convenient to use rescaled linear-operator-valued formal power series

$$\begin{aligned} {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z), {\mathcal {R}}_{\upsilon \phi }(z) \in {{\,\textrm{End}\,}}(W \otimes W)[[z]], \end{aligned}$$

(5.9)

uniquely defined by the condition that they fix $w_0 \otimes w_0$:

$$\begin{aligned} {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z) \,\propto & {} \, (\varrho \otimes {{\bar{\varrho }}}^{\, -})({\mathcal {R}}(z)), \qquad {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z) \cdot (w_0 \otimes w_0) = w_0 \otimes w_0, \nonumber \\ {\mathcal {R}}_{\upsilon \phi }(z) \,\propto & {} \, (\upsilon \otimes \phi ^-)({\mathcal {R}}(z)), \qquad {\mathcal {R}}_{\upsilon \phi }(z) \cdot (w_0 \otimes w_0) = w_0 \otimes w_0. \end{aligned}$$

(5.10)

These power series will appear in the boundary factorization identity. In Appendix B we obtain explicit expressions for ${\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z)$ and ${\mathcal {R}}_{\upsilon \phi }(z)$, although we will not need these for the proof of the boundary factorization identity using the universal K-matrix formalism of Sect. 3.

6 K-Operators

In this section we consider solutions of reflection equations associated to the subalgebra $U_q(\mathfrak {k})$.

6.1 Right K-operators

By Theorem 3.6, applying any of the level-0 $U_q({{\widehat{\mathfrak {b}}}}^+)$-representations $\varrho$, ${{\bar{\varrho }}}$, $\upsilon$, $\phi$ to the grading-shifted universal K-matrix associated to $U_q(\mathfrak {k})$ we obtain ${{\,\textrm{End}\,}}(W)$-valued formal power series, satisfying the reflection equation (3.7). Moreover, in terms of the terminology of Sect. 2.7, the weight subspaces of all four actions are all 1-dimensional and therefore $w_0$ is an eigenvector of each action with invertible eigenvalue. We will consider the scalar multiples of these linear operators which fix $w_0$:

$$\begin{aligned} {\mathcal {K}}_\pi (z) \, \propto \, \pi ({\mathcal {K}}(z)), \qquad {\mathcal {K}}_\pi (z) \cdot w_0 = w_0. \end{aligned}$$

(6.1)

for $\pi \in \{ \varrho , {{\bar{\varrho }}}, \upsilon , \phi \}$. It is convenient to obtain explicit expressions by applying Propositions 3.7 and 3.8. These could be found independently of the universal K-matrix formalism, either by solving the reflection equations directly in all cases or by following the approach outlined in [DG02, DM03, RV16] (this relies on the irreducibility of certain tensor products as $U_q(\mathfrak {k})((z))$-modules; otherwise the reflection equation must be verified directly).

First of all, the linear operator

$$\begin{aligned} K_{\Pi }(z) = \begin{pmatrix} \xi z^2 - 1 &{} 0 \\ 0 &{} \xi - z^2 \end{pmatrix} \in {{\,\textrm{End}\,}}({\mathbb {C}^2}) [[z]] \end{aligned}$$

(6.2)

is, up to a scalar, the unique solution of the $U_q(\mathfrak {k})$-intertwining condition

$$\begin{aligned} K_\Pi (z) \Pi _z(u) = \Pi _{1/z}(u) K_\Pi (z) \qquad \text {for all } u \in U_q(\mathfrak {k}). \end{aligned}$$

(6.3)

By Theorem 3.6, it is proportional to the action of the grading-shifted universal K-matrix in the representation $(\Pi ,\mathbb {C}^2)$.

Recall that $\Pi \circ \psi = \Pi$. Hence, motivated by Proposition 3.7, we consider the right reflection equation for $\pi \in \{\varrho , {{\bar{\varrho }}}, \upsilon , \phi \}$:

$$\begin{aligned} {\mathcal {L}}_\pi (\tfrac{y}{z}) {\mathcal {K}}_{\pi }(y) {\mathcal {L}}_{\pi }(yz) K_{\Pi }(z) = K_{\Pi }(z) {\mathcal {L}}_\pi (y z) {\mathcal {K}}_{\pi }(y) {\mathcal {L}}_{\pi }(\tfrac{y}{z}) \in {{\,\textrm{End}\,}}(W \otimes {\mathbb {C}^2})[[y/z,z]].\nonumber \\ \end{aligned}$$

(6.4)

Lemma 6.1

We have

$$\begin{aligned} {\mathcal {K}}_\varrho (z)= & {} (-q^{-D} \xi )^{D} (q^{2}\xi ^{-1}z^2;q^2)_{D}, \,\,\qquad {\mathcal {K}}_{{\bar{\varrho }}}(z) = (q z^2)^{-D} (q^{2} \xi ^{-1} z^{-2};q^2)_D^{-1}, \nonumber \\ {\mathcal {K}}_\upsilon (z)= & {} z^{-2D} \frac{(q^{2-\mu } \xi ^{-1} z^2;q^2)_D}{(q^{2-\mu } \xi ^{-1} z^{-2};q^2)_D}, \quad \qquad {\mathcal {K}}_\phi (z) \,= (-q^{-\mu - D-1} \; \xi )^D. \end{aligned}$$

(6.5)

Note that these expressions were already given in [BT18] in different conventions. For completeness we sketch a proof relying on the universal K-matrix formalism.

Proof of Lemma 6.1

For ${\mathcal {K}}_\upsilon (z)$, by a straightforward check, the intertwining condition

$$\begin{aligned} {\mathcal {K}}_\upsilon (z) \upsilon _{z}(u) = \upsilon _{1/z}(u) {\mathcal {K}}_\upsilon (z) \qquad \text {for all } u \in U_q(\mathfrak {k}) \end{aligned}$$

(6.6)

can be solved to find ${\mathcal {K}}_{\upsilon }(z)$, making use of Proposition 3.8. Since ${\mathcal {K}}(z)$ commutes with the action of $k_1$ it follows that ${\mathcal {K}}_\upsilon (z) = f(D)$ for some $f \in {\mathcal {F}}$. Now imposing (6.6) for the generators $e_0-q^{-1}\xi ^{-1}k_0f_1$ and $e_1-q^{-1}\xi k_1f_0$ yields the recurrence relation

$$\begin{aligned} \frac{f(D+1)}{f(D)} = \frac{1-q^{2(D+1)-\mu } \xi ^{-1} z^2}{z^2-q^{2(D+1)-\mu } \xi ^{-1}}. \end{aligned}$$

In particular, the linear relation (6.6) has a 1-dimensional solution space. Together with the constraint $f(0)=1$ it yields the formula given in (6.5).

For $\pi \in \{ \varrho , {{\bar{\varrho }}}, \phi \}$, it is convenient to consider the linear space

$$\begin{aligned} \textsf{RE}_\pi := \{ {\mathcal {K}}_\pi (y) \in {\mathcal {F}}(D) \, | \, (6.4) \text { is satisfied} \} \end{aligned}$$

(6.7)

and use Proposition 3.7 to find the explicit expression, relying on the second part of Theorem 3.6 for the fact that ${\mathcal {K}}_\pi (y)$ lies in ${\mathcal {F}}(D)$. Indeed, the operator $K_{\varrho }(z)$ was obtained in [VW20, Sec. 2.4] as the unique element of the 1-dimensional linear space $\textsf{RE}_{\varrho }$ which fixes $w_0$. In an analogous way we obtain the result for $K_{{{\bar{\varrho }}}}(z)$.

Note that $\phi$ is a reducible representation. Indeed, the solution space of (6.4) with $\pi =\phi$ is infinite-dimensional: the general solution ${\mathcal {K}}_\phi (z)$ is of the form $(- q^{-\mu - D -1} \; \xi )^D p$ with p in the centralizer of a in ${\mathcal {A}}$, i.e. a polynomial in a with coefficients in $\mathbb {C}[[z]]$. Since ${\mathcal {K}}_\phi (z) \in {\mathcal {F}}(D)$, p is a scalar. The requirement that $w_0$ is fixed forces $p=1$. $\square$

6.2 Left K-operators

We now obtain linear-operator-valued power series satisfying a reflection equation for the left boundary by using a well-established bijection, see [Sk88, Eq. (15)], between its solution set and the solution set of the right reflection equation. For fixed $\widetilde{\xi }\in \mathbb {C}^\times$ we define

$$\begin{aligned} {\widetilde{K}}_{\Pi }(z):= & {} (1-q^2 \widetilde{\xi }^{-1} z^2)^{-1} (1-q^2 \widetilde{\xi }z^2)^{-1} \big ( K_{\Pi }(q z)^{-1}|_{\xi \mapsto \widetilde{\xi }^{-1}} \big ) \nonumber \\= & {} \begin{pmatrix} q^2 \widetilde{\xi }z^2 -1 &{} 0\\ 0 &{} \widetilde{\xi }- q^2z^2 \end{pmatrix}. \end{aligned}$$

(6.8)

Also, for $\pi \in \{ \varrho , {{\bar{\varrho }}}, \upsilon , \phi \}$ we define

$$\begin{aligned} {\widetilde{{\mathcal {K}}}}_{\pi }(z):= {\mathcal {K}}_{\pi }(q z)^{-1}|_{\xi \mapsto \widetilde{\xi }^{-1}}. \end{aligned}$$

(6.9)

Similarly, note that ${\mathcal {L}}_\pi (\gamma z)$ is invertible in ${{\,\textrm{End}\,}}(W \otimes \mathbb {C}^2)[[z]]$ for all $\gamma \in \mathbb {C}$. We define

$$\begin{aligned} {\widetilde{{\mathcal {L}}}}_\pi (z) = {\mathcal {L}}_\pi (q^2z)^{-1}. \end{aligned}$$

(6.10)

Lemma 6.2

For all $\pi \in \{\varrho , {{\bar{\varrho }}}, \upsilon , \phi \}$ the left reflection equation holds:

$$\begin{aligned} {\widetilde{{\mathcal {K}}}}_\pi (y) {\widetilde{{\mathcal {L}}}}_\pi (yz) {\widetilde{K}}_{\Pi }(z) {\mathcal {L}}_\pi (\tfrac{y}{z}) \!=\! {\mathcal {L}}_\pi (\tfrac{y}{z}) {\widetilde{K}}_{\Pi }(z){\widetilde{{\mathcal {L}}}}_\pi (yz){\widetilde{{\mathcal {K}}}}_\pi (y) \quad \!\!\!\!\in \! {{\,\textrm{End}\,}}(W \otimes {\mathbb {C}^2})[[y/z,z]].\nonumber \\ \end{aligned}$$

(6.11)

Proof

The desired equation (6.11) can be rewritten as

$$\begin{aligned} {\widetilde{K}}_{\Pi }(z)^{-1}{\widetilde{{\mathcal {L}}}}_\pi (yz)^{-1} {\widetilde{{\mathcal {K}}}}_\pi (y)^{-1} {\mathcal {L}}_\pi (\tfrac{y}{z}) = {\mathcal {L}}_\pi (\tfrac{y}{z}) {\widetilde{{\mathcal {K}}}}_\pi (y)^{-1}{\widetilde{{\mathcal {L}}}}(yz)^{-1}{\widetilde{K}}_{\Pi }(z)^{-1}. \end{aligned}$$

By (6.8–6.10), this is equivalent to the right-reflection equation (6.4) with $y \mapsto qy$, $z \mapsto qz$ and $\xi \mapsto \widetilde{\xi }^{-1}$. $\square$

Using the explicit formulas (6.2) and (6.4) we obtain that the solutions of the left reflection equations (6.9) are the following ${{\,\textrm{End}\,}}(W)$-valued formal power series in z:

$$\begin{aligned} {\widetilde{{\mathcal {K}}}}_\varrho (z)= & {} (-q^D \widetilde{\xi })^D (q^4 \widetilde{\xi }z^2;q^2)_{D}^{-1}, \quad \qquad \quad \, {\widetilde{{\mathcal {K}}}}_{{\bar{\varrho }}}(z) = (q^3 z^2)^D (\widetilde{\xi }z^{-2};q^2)_{D}, \nonumber \\ {\widetilde{{\mathcal {K}}}}_{\upsilon }(z)= & {} (q z)^{2D} \frac{(q^{-\mu } \widetilde{\xi }z^{-2};q^2)_{D}}{(q^{4-\mu } \widetilde{\xi }z^2;q^2)_{D}}, \quad \quad \qquad {\widetilde{{\mathcal {K}}}}_{\phi }(z)\, = (-q^{\mu + D + 1} \widetilde{\xi })^D. \end{aligned}$$

(6.12)

7 Fusion Intertwiners Revisited

In this short intermezzo we explain how the universal K-matrix formalism naturally leads to relations involving K-operators and $U_q(\mathfrak {b}^+)$-intertwiners, called fusion intertwiners, which play a key role in the short exact sequence approach to the Q-operator. These intertwiners were discussed in [VW20] and the relevant relations with K-matrices were shown by a linear-algebraic computation relying on the explicit expressions of the various constituent factors, see [VW20, Lem. 3.2]. In other words, the representation-theoretic origin of these relations was unclear, which we now remedy.

Level-0 representations of $U_q({{\widehat{\mathfrak {b}}}}^+)$ are amenable to scalar modifications of the action of $U_q(\mathfrak {h}) = \langle k_1^{\pm 1} \rangle$, see also [HJ12, Rmk. 2.5]. In particular, for $r \in \mathbb {C}^\times$, define a modified Borel representation $\varrho$ as follows:

$$\begin{aligned} \varrho _r(e_i) = \varrho (e_i), \qquad \varrho _r(k_0) = r \varrho (k_0), \qquad \varrho _r(k_1) = r^{-1} \varrho (k_1) \end{aligned}$$

(7.1)

and consider the grading-shifted representation $\varrho _{r,z}:= (\varrho _r)_z$. There exist $U_q({{\widehat{\mathfrak {b}}}}^+)$-intertwiners

$$\begin{aligned} \iota (r)&: (\varrho _{qr,qz},W) \rightarrow (\varrho _{r,z} \otimes \Pi _z,W \otimes \mathbb {C}^2), \\ \tau (r)&: (\varrho _{r,z} \otimes \Pi _z,W \otimes \mathbb {C}^2) \rightarrow (\varrho _{q^{-1}r,q^{-1}z},W), \end{aligned}$$

called fusion intertwiners, which take part in the following short exact sequence:

Explicitly¹¹, we have

$$\begin{aligned} \iota (r) = \begin{pmatrix} q^{-D} a^\dag \\ -q^{D+1} r \end{pmatrix}, \qquad \tau (r) = \begin{pmatrix} q^D,&q^{-D} r^{-1} a^\dag \end{pmatrix}. \end{aligned}$$

(7.3)

Analogously to Theorem 5.2, fusion relations for the L-operators ${\mathcal {L}}(r,z)$, defined as suitable scalar multiples of $(\varrho _{r,z} \otimes \Pi )({\mathcal {R}})$, now follow from these intertwining properties and the coproduct formulas for ${\mathcal {R}}$ (2.16), see [VW20, Eqns. (3.8) and (3.9)].

Recalling the universal object ${\mathcal {K}}$ and Theorem 3.6, we define the corresponding K-operator ${\mathcal {K}}_\varrho (r,z)$ as the unique scalar multiple of $\varrho _{r,z}({\mathcal {K}})$ which fixes $w_0$ (cf. [VW20, Prop. 2.5]). Then

$$\begin{aligned} (\varrho _{r,z} \otimes \Pi _z)(\Delta ({\mathcal {K}})) \qquad \propto \qquad {\mathcal {K}}_\varrho (r,z)_1 {\mathcal {L}}(r,z^2) K_\Pi (z)_2 \end{aligned}$$

(7.4)

as a consequence of (3.19). Since ${\mathcal {K}}$ lies in a completion of $U_q({{\widehat{\mathfrak {b}}}}^+)$, the intertwining properties of $\iota (r)$ and $\tau (r)$ now directly yield the following fusion relation for the K-operator:

$$\begin{aligned} {\mathcal {K}}_\varrho (r,z)_1 {\mathcal {L}}(r,z^2) K_\Pi (z)_2 \iota (r) \qquad&\propto \qquad \iota (r) {\mathcal {K}}_\varrho (qr,qz) \\ \tau (r) {\mathcal {K}}_\varrho (r,z)_1 {\mathcal {L}}(r,z^2) K_\Pi (z)_2 \qquad&\propto \qquad {\mathcal {K}}_\varrho (q^{-1}r,q^{-1}z) \tau (r), \end{aligned}$$

with the scalar factors determined by applying the two sides of the equation to $w_0$, say. We will be able to prove a boundary counterpart of the factorization identity (5.7) using similar ideas.

We recover, with a much smaller computational burden, the key result [VW20, Lemma 3.2] (a similar relation for left K-operators can easily be deduced from this, as explained in the last sentence of [VW20, Proof of Lemma 3.2]). In the approach to Baxter’s Q-operator using short exact sequences, the fusion relations for L and K-operators induce fusion relations for 2-boundary monodromy operators, see [VW20, Lem. 4.2] from which Baxter’s relation (1.1) follows by taking traces, see [VW20, Sec. 5.2].

8 Boundary Factorization Identity

In motivating and presenting the key boundary relations, it is very useful to introduce a graphical representation of spaces and operators. Let us introduce the following pictures for the different representations from Sects. 4 and 5:

For any vector spaces V, $V'$, denote by ${\mathcal {P}}$ the linear map from $V \otimes V'$ to $V' \otimes V$ such that ${\mathcal {P}}(v \otimes v') = v' \otimes v$ for all $v \in V$, $v' \in V'$. Also set $z = z_1/z_2$. We then have the following pictures for L-operators and R-operators:

We now make the following definitions¹²:

$$\begin{aligned} \widetilde{{\mathcal {R}}}_{\varrho {{\bar{\varrho }}}}(z):= {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(q^2z)^{-1}, \qquad \widetilde{{\mathcal {R}}}_{\upsilon \phi }(z):= {\mathcal {R}}_{\upsilon \phi }(q^2z)^{-1}, \end{aligned}$$

(8.1)

and represent these modified R-matrices by the following pictures:

The various right-boundary K-matrices are represented as follows:

The left-boundary K-matrices defined in Sect. 6.2 are represented by the natural analogues of these pictures. For example:

Making use of these pictures, we see that Theorem 5.2 and Corollary 5.3 are represented by

For the compatibility with the right boundary we claim that

which corresponds to the following identity in ${\mathcal {A}}^{(2)}$:

$$\begin{aligned} {\mathcal {K}}_\upsilon (z)_1 {\mathcal {R}}_{\upsilon \phi }(z^2) {\mathcal {K}}_\phi (z)_2 \,{\mathcal {O}}= {\mathcal {O}}\, {\mathcal {K}}_\varrho (q^{-\mu /2}z)_1 {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z^2) {\mathcal {K}}_{{\bar{\varrho }}}(q^{\mu /2}z)_2, \end{aligned}$$

(8.2)

which we call the right boundary factorization identity. The diagrams above serve as a motivation for the identity, which we now prove using results from Sect. 3 (an alternative computational proof of Theorem 8.1 is given in Appendix C).

Theorem 8.1

For all $\mu \in \mathbb {C}$, all $q \in \mathbb {C}^\times$ not a root of unity and all $\xi \in \mathbb {C}^\times$, relation (8.2) is satisfied.

Proof

The proof is analogous to the proof of Theorem 5.2. We first note that

$$\begin{aligned} \begin{aligned} \big ( \varrho _{q^{-\mu /2}z} \otimes {{\bar{\varrho }}}_{q^{\mu /2}z} \big )\big ( (\textsf{id}\otimes \psi )({\mathcal {R}}) \big )&= \big ( \varrho _{q^{-\mu /2}z} \otimes {{\bar{\varrho }}}^{\, -}_{q^{-\mu /2}z^{-1}} \big )({\mathcal {R}}){} & {} \propto \; {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z^2), \\ \big ( \upsilon _{z} \otimes \phi _{z} \big )\big ( (\textsf{id}\otimes \psi )({\mathcal {R}}) \big )&= \big ( \upsilon _{z} \otimes \phi ^-_{z^{-1}} \big )({\mathcal {R}}){} & {} \propto \; {\mathcal {R}}_{\upsilon \phi }(z^2). \end{aligned} \end{aligned}$$

Noting the coproduct formula (3.19), we obtain

$$\begin{aligned} \begin{aligned} {\mathcal {K}}_\varrho (q^{-\mu /2}z)_1 {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z^2) {\mathcal {K}}_{{\bar{\varrho }}}(q^{\mu /2}z)_2 \quad&\propto \quad \big ( \varrho _{q^{-\mu /2}z} \otimes {{\bar{\varrho }}}_{q^{\mu /2}z} \big )(\Delta ({\mathcal {K}})), \\ {\mathcal {K}}_\upsilon (z)_1 {\mathcal {R}}_{\upsilon \phi }(z^2) {\mathcal {K}}_\phi (z)_2 \quad&\propto \quad \big ( \upsilon _{z} \otimes \phi _{z} \big )(\Delta ({\mathcal {K}})). \end{aligned} \end{aligned}$$

Now Theorem 4.4 implies (8.2) up to a scalar. The fact that all factors fix $w_0 \otimes w_0$ shows that the scalar is 1. $\square$

Compatibility with the left boundary requires that

The identity in ${\mathcal {A}}^{(2)}$ corresponding to this is

$$\begin{aligned} {\widetilde{{\mathcal {K}}}}_{{\bar{\varrho }}}(q^{\mu /2}z,{\widetilde{\xi }})_2 \widetilde{{\mathcal {R}}}_{\varrho {{\bar{\varrho }}}}(z^2) {\widetilde{{\mathcal {K}}}}_\varrho (q^{-\mu /2}z,{\widetilde{\xi }})_1 {\mathcal {O}}^{-1} = {\mathcal {O}}^{-1} {\widetilde{{\mathcal {K}}}}_\phi (z,{\widetilde{\xi }})_2 \widetilde{{\mathcal {R}}}_{\upsilon \phi }(z^{2}) {\widetilde{{\mathcal {K}}}}_\upsilon (z,{\widetilde{\xi }})_1.\nonumber \\ \end{aligned}$$

(8.3)

Theorem 8.2

Relation (8.3) is satisfied.

Proof

Given the definitions (6.12) and (8.1), this follows straightforwardly by inverting (8.2) and replacing $(z,\xi ) \mapsto (qz,\widetilde{\xi }^{-1})$. $\square$

9 Discussion

The main result of this paper is Theorem 8.1 which can be viewed as a boundary analogue of Theorem 5.2. To establish this result, first we needed to show that all R and K-operators involved in equation (8.2) are well-defined actions of the universal elements ${\mathcal {R}}$ and ${\mathcal {K}}$ on the infinite-dimensional $U_q({{\widehat{\mathfrak {b}}}}^+)$-modules involved. The key fact that allows for this is that ${\mathcal {R}}$ and ${\mathcal {K}}$ live in completions of $U_q({{\widehat{\mathfrak {b}}}}^+)\otimes U_q({{\widehat{\mathfrak {b}}}}^-)$ and of $U_q({{\widehat{\mathfrak {b}}}}^+)$, respectively. This is very familiar for ${\mathcal {R}}$ but for ${\mathcal {K}}$ relies on the recent works [AV22a, AV22b]. Introducing the $U_q({{\widehat{\mathfrak {b}}}}^+)$-intertwiner ${\mathcal {O}}$ and the formula for $\Delta ({\mathcal {K}})$ given by (3.19), relation (8.2) follows immediately from the intertwining property of ${\mathcal {O}}$.

The open Q-operator ${\mathcal {Q}}(z)$ of [VW20] is the trace of a product of R and K-operators over the $U_q({{\widehat{\mathfrak {b}}}}^+)$-module $(\varrho _z,W)$ and there is a similar construction of an open Q-operator $\overline{{\mathcal {Q}}}(z)$. In a future paper, the authors will present this construction and the use of Theorem 8.2 in deriving a boundary analogue of the factorization relation $\mathcal T_{\mu }(z) \, \propto \, \mathcal Q(zq^{-\mu /2}) \overline{\mathcal Q}(zq^{\mu /2})$. They will also develop the analogous theory for different coideal subalgebras, in particular those for which non-diagonal solutions of the reflection equation are intertwiners. There is a quite subtle rational degeneration of the construction in the present paper. The first-named author will study this in a separate paper, giving an alternative approach to Q-operators for the open XXX spin chain, cf. [FS15].

Appendices

Deformed Pochhammer Symbols and Exponentials

This appendix is independent from the main text, but provides identities which are used there. We review some basic theory of deformed Pochhammer symbols and exponentials (as formal power series) with a deformation parameter $p \in \mathbb {C}^\times$, which corresponds to $q^2$ in the main text.

1.1 Deformed Pochhammer symbols

Let x be a formal variable. For $n \in \mathbb {Z}$, the (finite) deformed Pochhammer symbol $(x;p)_n \in \mathbb {C}[[x]]$ is defined by

$$\begin{aligned} (x;p)_n:= {\left\{ \begin{array}{ll} \displaystyle \prod _{m=0}^{n-1} (1-x p^m) &{} \text {if } n \geqslant 0, \\ \displaystyle \prod _{m=n}^{-1} (1-x p^m)^{-1} &{} \text {if } n <0 \end{array}\right. } \end{aligned}$$

(A.1)

(the definition for $n<0$ is understood as a product of geometric series); since its constant coefficient is nonzero, it is invertible. For all $p \in \mathbb {C}^\times$ and $n \in \mathbb {Z}_{\geqslant 0}$ we have the following basic identity in $\mathbb {C}[[x]]$, see [GR90, (I.2), (I.3)]:

$$\begin{aligned} (x;p)_{-n} = (p^{-n} x;p)_n^{-1} = (x/p;p^{-1})_n^{-1} = (-x)^{-n} p^{n(n+1)/2} (p/x;p)_n^{-1}. \end{aligned}$$

(A.2)

Assuming $|p|<1$, the infinite deformed Pochhammer symbol

$$\begin{aligned} (x;p)_\infty := \prod _{m=0}^\infty (1-x p^m) \end{aligned}$$

(A.3)

is an invertible formal power series with well-defined coefficients in $\mathbb {C}$. The following identity holds in $\mathbb {C}[[x]]$, see [GR90, (I.5)]:

$$\begin{aligned} (x;p)_n = \frac{(x; p)_\infty }{(p^n x; p)_\infty }. \end{aligned}$$

(A.4)

1.2 Deformed exponentials

From now on we assume that p is not a root of unity. In particular, $(p;p)_k \ne 0$ for all $k \in \mathbb {Z}_{\geqslant 0}$. The deformed exponential is the invertible formal power series

$$\begin{aligned} e_{p}(x):= {}_1\phi _0(0;-;p,x) = \sum _{k=0}^\infty \frac{x^k}{(p;p)_k}. \end{aligned}$$

(A.5)

The ordinary exponential formal power series arises as the termwise limit $\lim _{p \rightarrow 1} e_p((1-p)x) = {\mathrm e}^x$. This series satisfies the functional relation

$$\begin{aligned} e_{p}(p x) = (1-x) e_{p}(x), \end{aligned}$$

(A.6)

see [GR90, Sec. 1.3]. Since constants are the only formal power series which are invariant under $x \mapsto px$, an inspection of constant coefficients shows that (A.6) implies

$$\begin{aligned} e_{p}(x) = \frac{1}{(x;p)_\infty } \qquad \text {if } |p|<1. \end{aligned}$$

(A.7)

Similarly we consider the invertible formal power series

$$\begin{aligned} E_p(x):= {}_0\phi _0(-;-;p,-x) = \sum _{k=0}^\infty \frac{p^{k(k-1)/2} x^k}{(p;p)_k}. \end{aligned}$$

(A.8)

Then $E_p(-x)^{-1}$ also satisfies (A.6) and by comparing constant coefficients again we deduce $e_{p}(x) = E_p(-x)^{-1}$. By evaluating (A.2) at $x=1$, we obtain $E_p(-x) = e_{p^{-1}}(p^{-1}x)$ and hence

$$\begin{aligned} e_{p}(x) = e_{p^{-1}}(p^{-1}x)^{-1} \quad \in \mathbb {C}[[x]]. \end{aligned}$$

(A.9)

Deformed exponentials in x and y satisfy various useful identities in particular deformations of the commutative algebra $\mathbb {C}[[x,y]]$. For instance, in any algebra generated by the symbols x and y such that $yx = \gamma xy$ for $\gamma \in \mathbb {C}$, the definition implies the following identity:

$$\begin{aligned} y e_p(x) = e_p(\gamma x) y \end{aligned}$$

(A.10)

which we will use repeatedly. For a survey of product formulas analogous to $\exp (x)\exp (y)=\exp (x+y)$, see [Ko97]. We will need the following result.

Lemma A.1

Let x, y be elements of an algebra such that $y x = p x y$. The following identities hold as formal power series in x, y:

$$\begin{aligned} e_p(x) e_p(y)&= e_p(x+y), \end{aligned}$$

(A.11)

$$\begin{aligned} e_p(y) e_p(x)&= e_p\big (x(1-y)\big ) e_p(y) \nonumber \\ {}&= e_p(x) e_p(-xy) e_p(y) = e_p(x) e_p\big ((1-x)y\big ). \end{aligned}$$

(A.12)

Proof

(A.11) is a direct consequence of the well-known q-binomial formula, see e.g. [GR90, Ex. 1.35]. For (A.12) see [Ko97, Prop. 3.2]. $\square$

1.3 Deformed exponentials as linear maps

Let V be a $\mathbb {C}$-linear space. Call an operator f on V locally nilpotent if for all $v \in V$ there exists $o(v) \in \mathbb {Z}_{\geqslant 0}$ such that $f^{o(v)}(v)=0$ (note that nilpotent operators are locally nilpotent and if V is finite-dimensional the converse is true). If f is nilpotent, the deformed exponential $e_p(f)$ defines an invertible map on V. If additionally y is an indeterminate then $e_p(yf)$ is a well-defined invertible element of ${{\,\textrm{End}\,}}(V)[[y]]$.

In the case $V=W \otimes W$ the following commutation relations for linear-operator valued formal series are satisfied, expressed in terms of the linear operators a, $a^\dag$, ${{\bar{a}}}^\dag$, f(D) ($f \in {\mathcal {F}}$) on W defined in Sect. 4.2.

Lemma A.2

Let y be a formal variable. In ${{\,\textrm{End}\,}}(W \otimes W)[[y]]$ the following identities hold:

$$\begin{aligned} \big [ e_p(y a_1 {{\bar{a}}}^\dag _2), f(D_1+D_2) \big ] = \big [ e_p(y a_1 {{\bar{a}}}^\dag _2), a_1 \big ] = \big [ e_p(y a_1 {{\bar{a}}}^\dag _2), {{\bar{a}}}^\dag _2 \big ] = 0 \end{aligned}$$

(A.13)

for all $f \in {\mathcal {F}}$ and

$$\begin{aligned} \big [ e_p(y a_1 {{\bar{a}}}^\dag _2), a^\dag _1 \big ]&= y p^{D_1} {{\bar{a}}}^\dag _2 e_p(y a_1 {{\bar{a}}}^\dag _2), \end{aligned}$$

(A.14)

$$\begin{aligned} \big [ e_p(y a_1 {{\bar{a}}}^\dag _2), p^{-D_1} a_2 \big ]&= y e_p(y a_1 {{\bar{a}}}^\dag _2) a_1 p^{-D_1}. \end{aligned}$$

(A.15)

Proof

Note that (A.13) follows directly from the definition of the deformed exponential. A straightforward inductive argument using (4.4) yields

$$\begin{aligned}{}[a^{k+1},a^\dag ]&= (1-p^{k+1}) p^D a^k, \end{aligned}$$

(A.16)

$$\begin{aligned} _{p^{k+1}}&= (1-p^{k+1}) ({{\bar{a}}}^{\dag })^k, \end{aligned}$$

(A.17)

for all $k \in \mathbb {Z}_{\geqslant 0}$, which imply (A.14) and (A.15), respectively. $\square$

Explicit Expressions for R-Operators

In this appendix we derive explicit formulas for ${\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z)$ and ${\mathcal {R}}_{\upsilon \phi }(z)$, defined by (5.10) as images of the universal R-matrix ${\mathcal {R}}$ fixing $w_0 \otimes w_0$. We expect that these will be useful in further studies of Baxter’s Q-operators for the open XXZ spin chain; for now they will allow us to give a proof of the boundary factorization identity which does not rely on the universal K-matrix formalism. First we note that, by the second part of Theorem 2.4, ${\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z)$ and ${\mathcal {R}}_{\upsilon \phi }(z)$ lie in the centralizer

$$\begin{aligned} {\mathcal {A}}^{(2)}_0:= \Big \{ X \in {\mathcal {A}}^{(2)} \, \Big | \, \big [ X, q^{D_1+D_2} \big ] = 0 \Big \}. \end{aligned}$$

(B.1)

One straightforwardly verifies that ${\mathcal {A}}^{(2)}_0$ is generated by elements of the form

$$\begin{aligned} \sum _{k \geqslant 0} ({{\bar{a}}}^\dag _2)^k f_k(D_1,D_2) a_1^k, \qquad \sum _{k \geqslant 0} (a^\dag _1)^k f_k(D_1,D_2) a_2^k, \qquad \qquad f_k \in {\mathcal {F}}^{(2)}. \end{aligned}$$

(B.2)

Hence, elements of ${\mathcal {A}}^{(2)}_0$ in fact commute with all elements of the form $f(D_1+D_2)$ ($f \in {\mathcal {F}}$).

1.1 Explicit expression for $R_{\upsilon \phi }(z)$

We first state and prove an explicit formula for $R_{\upsilon \phi }(z)$. We keep using the shorthand notation $p=q^2$.

Theorem B.1

For all $z \in \mathbb {C}$ we have

$$\begin{aligned} {\mathcal {R}}_{\upsilon \phi } (z) = e_p(z a^\dag _1 a_2) q^{(\mu -1)(D_2-D_1)-2 D_1 (D_2+1)}. \end{aligned}$$

(B.3)

Proof

From Proposition 2.6 we deduce that ${\mathcal {R}}_{\upsilon \phi } (z)$ is a solution of the linear relation

$$\begin{aligned} X (\upsilon _{z} \otimes \phi ^-)(\Delta (u)) = (\upsilon _{z} \otimes \phi ^-)(\Delta ^{\textrm{op}}(u)) X \quad \text {for all } u \in U_q({{\widehat{\mathfrak {b}}}}^-). \end{aligned}$$

(B.4)

First of all, note that the element in the right-hand side of (B.3) satisfies (B.4) with $u \in \{ k_0,k_1 \}$ and so it suffices to prove that the vector space

$$\begin{aligned} \mathcal X = \Big \{ X \in {\mathcal {A}}^{(2)}_0 \, \Big | \, X \text { satisfies } (B.4) \text { for } u \in \{ f_0,f_1\} \Big \} \end{aligned}$$

(B.5)

is spanned by $e_p(z^2 a^\dag _1 a_2) q^{(\mu -1)(D_2-D_1)-2 D_1 (D_2+1)}$. Using the explicit formulas (2.2), (4.8) and (4.16), we obtain that (B.4) is equivalent to the system

$$\begin{aligned}&X \Big ( z^{-1} a_1 (q^{-\mu }-q^{\mu -2D_1} ) q^{-\mu -2D_2-1} + q^{-1} a_2 \Big ) \\&\quad = \Big ( z^{-1} a_1 (q^{-\mu }-q^{\mu -2D_1} ) + q^{\mu -2(D_1+1)} a_2 \Big ) X, \\&\qquad X a^\dag _1 q^{\mu +1+2D_2} = a^\dag _1 X. \end{aligned}$$

Without loss of generality we may write $X = {\widetilde{X}} q^{(\mu -1)(D_2-D_1)-2 D_1 (D_2+1)}$ with ${\widetilde{X}} \in {\mathcal {A}}^{(2)}_0$. Hence (B.4) is equivalent to

$$\begin{aligned} z^{-1} [{\widetilde{X}},a_1 (1-p^{\mu - D_1} ) ] = p^{\mu -D_1- 1} a_2 {\widetilde{X}} - {\widetilde{X}} p^{D_1} a_2, \qquad \qquad [{\widetilde{X}}, a^\dag _1] = 0. \end{aligned}$$

(B.6)

It is straightforward to check that the centralizer in ${\mathcal {A}}^{(2)}_0$ of $a^\dag _1$ is the subalgebra generated by elements of the form $\sum _{k \geqslant 0} (a^\dag _1)^k f_k(D_2) a_2^k$ with $f_k \in {\mathcal {F}}$. It follows that ${\widetilde{X}}$ is of this form. Therefore (B.4) is equivalent to the single equation

$$\begin{aligned}{} & {} \sum _{k \geqslant 0} \big [(a^\dag _1)^k,a_1 (1-p^{\mu -D_1} ) \big ] f_k(D_2) a_2^k =\\{} & {} \quad = z \sum _{k \geqslant 0} (a^\dag _1)^k \big ( p^{\mu -D_1-k-1} f_k(D_2+1) - p^{D_1} f_k(D_2) \big ) a_2^{k+1}. \end{aligned}$$

The commutator vanishes if $k=0$ so in the left-hand side we replace k by $k+1$. For $k \geqslant 0$ we have

$$\begin{aligned} \big [(a^\dag )^{k+1},a (1-p^{\mu -D} ) \big ]&= (a^\dag )^k (1-p^{k+1}) (p^{\mu - D - k - 1} - p^D). \end{aligned}$$

Hence (B.4) is equivalent to the recurrence relation

$$\begin{aligned} (1-p^{k+1}) \big ( p^{\mu -D_1-k-1} - p^{D_1} \big ) f_{k+1}(D_2) = z \big ( p^{\mu -D_1-k-1} f_k(D_2+1) - p^{D_1} f_k(D_2) \big ). \end{aligned}$$

Viewing ${\mathcal {F}}^{(2)}(D_1,D_2)$ as an ${\mathcal {F}}(D_2)$-module, the elements $p^{\pm D_1}$ are linearly independent. Hence the above recurrence relation is equivalent to the system

$$\begin{aligned} (1-p^{k+1}) f_{k+1}(D) = z f_k(D + 1), \qquad f_k(D+1) = f_k(D). \end{aligned}$$

This is in turn equivalent to $f_k(D) \in (p;p)_k^{-1} z^k \mathbb {C}$ for $k \in \mathbb {Z}_{>0}$, as required. $\square$

1.2 The automorphism $\chi$ and the q-oscillator subalgebra $\widetilde{{\mathcal {A}}}$

To obtain an expression for $R_{\upsilon \phi }(z)$ in terms of deformed exponentials, it is very convenient to point out an additional automorphism $\chi$. It cannot be defined on all of ${\mathcal {A}}$ so we will consider a subalgebra $\widetilde{{\mathcal {A}}}$. First, consider the subalgebra $\widetilde{{\mathcal {F}}}(D) \subset {\mathcal {F}}(D)$ generated by

$$\begin{aligned} p^{\pm D(D+1)/2}, \qquad \gamma ^D, \qquad (p {\widetilde{\gamma }};p)_D^{\pm 1}, \qquad (p \gamma z^2;p)_D, \qquad (-\gamma z^2)^{-D} (p \gamma ^{-1} z^{-2};p)_D^{-1} \end{aligned}$$

for all $\gamma \in \mathbb {C}^\times$ and ${\widetilde{\gamma }} \in \mathbb {C}^\times \backslash p^{\mathbb {Z}}$.

For elements of $\widetilde{\mathcal F}(D)$, unlike general elements of $\mathcal F(D)$, the symbol D can be formally evaluated at negative integers. Accordingly, we define an involutive automorphism $\chi$ of $\widetilde{{\mathcal {F}}}(D)$ accomplishing the formal replacement $D \mapsto -D-1$. To be more precise, we set

$$\begin{aligned} \chi \big ( p^{\pm D(D+1)/2} \big )= & {} p^{\pm D(D+1)/2}, \qquad \qquad \chi \big (\gamma ^D \big ) = \gamma ^{-D-1}, \qquad \qquad \nonumber \\ \chi \big ( (p {\widetilde{\gamma }};p)_D^{\pm 1} \big )= & {} (1 - {\widetilde{\gamma }})^{\mp 1} p^{\pm D(D+1)/2} (-{\widetilde{\gamma }})^{\mp D} (p {\widetilde{\gamma }}^{-1};p)_D^{\mp 1}, \nonumber \\ \chi \big ( (p \gamma z^2;p)_D \big )= & {} (1 - \gamma z^2)^{-1} p^{D(D+1)/2} (-\gamma z^2)^{-D} (p \gamma ^{-1} z^{-2};p)_D^{-1}, \nonumber \\ \chi \big ( (-\gamma z^2)^{-D} (p \gamma ^{-1} z^{-2};p)_D^{-1} \big )= & {} (1 - \gamma z^2) p^{-D(D+1)/2} (p \gamma z^2;p)_D. \end{aligned}$$

(B.7)

We denote the subalgebra of ${{\,\textrm{End}\,}}(W)$ generated by $a^\dag$, a and $\widetilde{\mathcal F}(D)$ by $\widetilde{{\mathcal {A}}}$. It is straightforward to check that $\chi$ extends to a (non-involutive) algebra automorphism of $\widetilde{{\mathcal {A}}}$ by means of the assignments

$$\begin{aligned} \chi (a) = {{\bar{a}}}^\dag , \qquad \chi (a^\dag ) = a. \end{aligned}$$

(B.8)

We can formulate a completion of the tensor product $\widetilde{{\mathcal {A}}} \otimes \widetilde{{\mathcal {A}}}$ in a similar way as for ${\mathcal {A}}\otimes {\mathcal {A}}$. More precisely, we consider the subalgebra $\widetilde{{\mathcal {F}}}^{(2)}$ of ${\mathcal {F}}^{(2)}$ generated by the subsets $\widetilde{{\mathcal {F}}}(D_1)$, $\widetilde{{\mathcal {F}}}(D_2)$ and the special elements $p^{\pm D_1(D_2+1)}$. The completed tensorial square of $\widetilde{{\mathcal {A}}}$ is defined to be the subalgebra $\widetilde{{\mathcal {A}}}^{(2)}$ of ${{\,\textrm{End}\,}}(W \otimes W)$ generated by the elements (4.7) with $g_{k,\ell }, \, h_{k,\ell } \in \widetilde{{\mathcal {F}}}^{(2)}$. Note that the boundary factorization identity (8.2) is an identity in the subalgebra $\widetilde{{\mathcal {A}}}^{(2)} \subset {{\,\textrm{End}\,}}(W \otimes W)[[z]]$.

The automorphism

$$\begin{aligned} \chi ^{(2)}:= \sigma \circ (\chi \otimes \chi ^{-1}) \end{aligned}$$

(B.9)

of $\widetilde{{\mathcal {A}}} \otimes \widetilde{{\mathcal {A}}}$ naturally extends to an automorphism of $\widetilde{{\mathcal {A}}}^{(2)}$, fixing $p^{\pm D_1(D_2+1)}$ and acting termwise on power series in locally nilpotent operators.

Remark B.2

The map $\chi$ can be seen as an infinite-dimensional version of conjugation by anti-diagonal matrices; certain $U_q({{\widehat{\mathfrak {b}}}}^+)$-representations are naturally related this way. For instance, for the 2-dimensional representation $\Pi$, note that ${{\,\textrm{Ad}\,}}(J) \circ \Pi = \Pi \circ \Phi$ where ${{\,\textrm{Ad}\,}}$ denotes ‘conjugation by’ and $J = \big ( {\begin{matrix} 0 &{} 1 \\ 1 &{} 0 \end{matrix}} \big )$. In the same way, $\chi$ relates the prefundamental representations $\varrho$ and ${{\bar{\varrho }}}$ up to a twist by the diagram automorphism $\Phi$: $\chi \circ \varrho = {{\bar{\varrho }}}\circ \Phi$. Hence, the condition (2.19) and the 1-dimensionality of the solution space of the relevant linear equation implies $({{\,\textrm{Ad}\,}}(J) \otimes \chi ) ( {\mathcal {L}}_\varrho (z) ) = {\mathcal {L}}_{{\bar{\varrho }}}(z)$. At the same time, a suitable scalar multiple of ${\mathcal {R}}_{\Pi \, \Pi }(z)$, i.e. the R-matrix for the XXZ chain, is fixed by ${{\,\textrm{Ad}\,}}(J \otimes J)$ and we will see in Sect. B.3 that the same statement is true for ${\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z)$ and $\chi ^{(2)}$.

From (3.5) it follows that $\Phi (U_q(\mathfrak {k})) = U_q(\mathfrak {k})|_{\xi \mapsto \xi ^{-1}}$. Hence, the boundary counterparts of the above relations also involve inversion of the free parameter $\xi$:

$$\begin{aligned} {{\,\textrm{Ad}\,}}(J)\big ( K_{\Pi }(z) \big )|_{\xi \mapsto \xi ^{-1}} = -\xi \, K_{\Pi }(z), \qquad \chi ({\mathcal {K}}_\varrho (z))|_{\xi \mapsto \xi ^{-1}} = q^{-1} (z^{2} - \xi ^{-1})^{-1} {\mathcal {K}}_{{\bar{\varrho }}}(z). \end{aligned}$$

In fact, applying $\chi \otimes {{\,\textrm{Ad}\,}}(J)$ to the reflection equation (6.4) with $\pi = \varrho$ and inverting $\xi$ we see that

$$\begin{aligned} {\mathcal {K}}_\varrho (z) \mapsto \chi ({\mathcal {K}}_\varrho (z))|_{\xi \mapsto \xi ^{-1}} \end{aligned}$$

defines a bijection: $\textsf{RE}_\varrho \rightarrow \textsf{RE}_{{\bar{\varrho }}}$ of the solution spaces defined in (6.7).

We can use the map $\chi ^{(2)}$ to generate further relations similar to those in Lemma A.2.

Lemma B.3

Let y be a formal parameter. In ${{\,\textrm{End}\,}}(W \otimes W)[[y]]$ the following identities hold:

$$\begin{aligned}{}[{{\bar{a}}}^\dag _2,e_p(y a^\dag _1 a_2)]&= y e_p(y a^\dag _1 a_2) a^\dag _1 p^{-D_2-1}, \end{aligned}$$

(B.10)

$$\begin{aligned}{}[{{\bar{a}}}^\dag _1 a_2,e_p(y a_1 {{\bar{a}}}^\dag _2) ]&= y \big ( e_p(y a_1 {{\bar{a}}}^\dag _2) p^{-D_1-1} - p^{-D_2-1} e_p(y a_1 {{\bar{a}}}^\dag _2) \big ). \end{aligned}$$

(B.11)

Proof

In this proof we view the algebra ${\mathcal {A}}$ as a subalgebra of ${{\,\textrm{End}\,}}(W)[[y]]$, and similarly for ${\mathcal {A}}^{(2)}$. To prove (B.10), first we apply $\chi ^{(2)}$ to (A.14), obtaining

$$\begin{aligned}{}[e_p(y a_1 {{\bar{a}}}^\dag _2 ),a_2] = y a_1 p^{-D_2-1} e_p(y a_1 {{\bar{a}}}^\dag _2). \end{aligned}$$

(B.12)

Now consider the unique involutive algebra anti-automorphism $\eta : {\mathcal {A}}\rightarrow {\mathcal {A}}$ which exchanges a and $a^\dag$ and fixes f(D) for all $f \in {\mathcal {F}}$ and the unique involutive algebra anti-automorphism $\overline{\eta }: {\mathcal {A}}\rightarrow {\mathcal {A}}$ which exchanges a and ${{\bar{a}}}^\dag$ and fixes f(D) for all $f \in {\mathcal {F}}$. Then $\eta ^{(2)}:= \eta \otimes \overline{\eta }$ is an algebra antiautomorphism of ${\mathcal {A}}\otimes {\mathcal {A}}$. It extends in a natural way to an algebra antiautomorphism of ${\mathcal {A}}^{(2)}$. By applying $\eta ^{(2)}$ to (B.12) we obtain (B.10).

Finally, to prove (B.11), upon right-multiplying (A.15) by $p^{D_1+D_2+1}$ we obtain

$$\begin{aligned}{}[e_p(y a_1 {{\bar{a}}}^\dag _2), a_1 p^{D_2}] = y e_p(y a_1 {{\bar{a}}}^\dag _2) a_1 p^{D_2}. \end{aligned}$$

(B.13)

From (A.14) and (B.13) it follows that

$$\begin{aligned} {[}e_p(y a_1 {{\bar{a}}}^\dag _2), a^\dag _1 a_2 p^{D_2}]= & {} y \Big ( {{\bar{a}}}^\dag _2 p^{D_1} e_p(y a_1 {{\bar{a}}}^\dag _2) a_2 + a^\dag _1 e_p(y a_1 {{\bar{a}}}^\dag _2) a_1 \Big ) p^{D_2} \nonumber \\= & {} y \Big ( p^{D_1} e_p(y a_1 {{\bar{a}}}^\dag _2) \big (p^{D_2}-1\big ) + \big (1-p^{D_1}\big ) e_p(y a_1 {{\bar{a}}}^\dag _2) p^{D_2} \Big ) \nonumber \\= & {} y \big ( e_p(y a_1 {{\bar{a}}}^\dag _2) p^{D_2} - p^{D_1} e_p(y a_1 {{\bar{a}}}^\dag _2) \big ). \end{aligned}$$

(B.14)

Now (B.11) follows as the $\chi ^{(2)}$-image of (B.14). $\square$

1.3 Explicit expression for ${\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z)$

To aid the computation of ${\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z)$, consider the subalgebra $\widetilde{{\mathcal {A}}}^{(2)}_0 = \widetilde{{\mathcal {A}}}^{(2)} \cap {\mathcal {A}}^{(2)}_0$, which is preserved by $\chi ^{(2)}$.

Lemma B.4

${\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z)$ is a $\widetilde{{\mathcal {A}}}^{(2)}_0$-valued formal power series whose coefficients are fixed by $\chi ^{(2)}$.

Proof

It is clear from (4.10) and (4.16) that $\varrho \otimes {{\bar{\varrho }}}^{\, -}$ takes values in $\widetilde{{\mathcal {A}}} \otimes \widetilde{{\mathcal {A}}} \subset \widetilde{{\mathcal {A}}}^{(2)}$. Now recall (2.20) and note that the factor $\kappa$ acts as $p^{D_1(D_2+1)}$. Furthermore, noting the form of $(\Sigma _z \otimes \textsf{id})(\Theta )$ given by (2.26) with the components $\Theta _\lambda$ lying in $U_q({{\widehat{\mathfrak {n}}}}^+)_\lambda \otimes U_q({{\widehat{\mathfrak {n}}}}^+)_{-\lambda }$ ($\lambda \in {\widehat{Q}}^+$), we obtain that the action of $\mathcal {R}(z)$ on $(\varrho \otimes {{\bar{\varrho }}}^{\, -},W \otimes W)$ is by an element of $\widetilde{{\mathcal {A}}}^{(2)}_0$. For the second part, note that

$$\begin{aligned} \chi ^{(2)} \circ (\varrho \otimes {{\bar{\varrho }}}^{\, -}) = (\chi ^{-1} \otimes \chi ) \circ ({{\bar{\varrho }}}^{\, -} \otimes \varrho ) \circ \sigma = (\varrho \otimes {{\bar{\varrho }}}^{\, -}) \circ (\omega \otimes \omega ) \circ \sigma . \end{aligned}$$

Applying this to ${\mathcal {R}}(z)$, making use of (2.27), (2.24) and (2.18), we obtain $\chi ^{(2)}({\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z)) = {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z)$. $\square$

In the derivation of the formula for ${\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z)$, we rely on the following result.

Lemma B.5

The centralizer of the subset $\{ a^\dag _1, {{\bar{a}}}^\dag _2 \}$ in ${\mathcal {A}}^{(2)}$ is equal to $\mathbb {C}[[z]]$.

Proof

This centralizer is the intersection of the centralizer of $a^\dag _1$ and the centralizer of ${{\bar{a}}}^\dag _2$, which are easily found to be equal to

$$\begin{aligned} \bigg \{ \sum _{k,\ell \geqslant 0} (a^\dag _1)^k f_{k,\ell }(D_2) a_2^\ell \, \bigg | \, f_{k,\ell } \in {\mathcal {F}}\bigg \}, \qquad \bigg \{ \sum _{k,\ell \geqslant 0} ({{\bar{a}}}^\dag _2)^k g_{k,\ell }(D_1) a_1^\ell \, \bigg | \, g_{k,\ell } \in {\mathcal {F}}\bigg \}, \end{aligned}$$

respectively. Clearly their intersection is trivial. $\square$

Now we are ready to state and prove a formula for ${\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z)$ in terms of deformed exponentials.

Theorem B.6

For all z we have

$$\begin{aligned} {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z) = e_{q^2}(q^3 z a_1 {{\bar{a}}}^\dag _2) e_{q^2}(q^{-1} z a^\dag _1 a_2) q^{-2D_1(D_2+1)}. \end{aligned}$$

(B.15)

Proof

Clearly, $w_0 \otimes w_0$ is fixed by the expression on the right-hand side of (B.15). In the following we initially work over the ring $\mathbb {C}[[z,z_2]]$ for some new indeterminate $z_2$ and write $z_1 = z z_2$. By applying $\varrho _{z_1} \otimes \Pi _1 \otimes {{\bar{\varrho }}}^{\, -}_{z_2}$ to (2.17) and left and right-multiplying by ${\mathcal {L}}^-_{{{\bar{\varrho }}},23}(z_2^{-1})^{-1}$ we obtain

$$\begin{aligned} {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z)_{12} {\mathcal {L}}_\varrho (z_1)_{13} {\mathcal {L}}^-_{{{\bar{\varrho }}}}(z_2^{-1})^{-1}_{32} = {\mathcal {L}}^-_{{{\bar{\varrho }}}}(z_2^{-1})^{-1}_{32} {\mathcal {L}}_\varrho (z_1)_{13} {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z)_{12} \end{aligned}$$

(B.16)

an equation in $(\widetilde{{\mathcal {A}}}^{(2)} \otimes {{\,\textrm{End}\,}}({\mathbb {C}^2}))[[z_2]]$. By a direct computation we obtain

$$\begin{aligned} {\mathcal {L}}^-_{{\bar{\varrho }}}(z_2^{-1})^{-1} = \frac{1}{z_2^2-1} \begin{pmatrix} q^{-D-1} z_2^2 &{} {{\bar{a}}}^\dag q^{-D-1} z_2 \\ a q^{D-1} z_2 &{} q^{D+1} z_2^2 - q^{-D-1} \end{pmatrix} \in {{\,\textrm{End}\,}}({\mathbb {C}^2}) \otimes \widetilde{{\mathcal {A}}}. \end{aligned}$$

(B.17)

Now we consider the equation

$$\begin{aligned} (z_2^2-1) X_{12} {\mathcal {L}}_\varrho (z_1)_{13} {\mathcal {L}}^-_{{{\bar{\varrho }}}}(z_2^{-1})^{-1}_{32} = (z_2^2-1) {\mathcal {L}}^-_{{{\bar{\varrho }}}}(z_2^{-1})^{-1}_{32} {\mathcal {L}}_\varrho (z_1)_{13} X_{12} \end{aligned}$$

(B.18)

in $(\widetilde{{\mathcal {A}}}^{(2)} \otimes {{\,\textrm{End}\,}}({\mathbb {C}^2}))[[z_2]]$, for some $X \in \widetilde{{\mathcal {A}}}^{(2)}_0$ such that $\chi ^{(2)}(X)=X$. It suffices to prove that

$$\begin{aligned} \mathcal X = \Big \{ X \in \widetilde{{\mathcal {A}}}^{(2)}_0 \, \Big | \, X \text { satisfies } (B.18) \text { and is fixed by } \chi ^{(2)} \Big \}, \end{aligned}$$

(B.19)

which by Lemma B.4 contains $(\varrho _z \otimes {{\bar{\varrho }}}^{\, -})({\mathcal {R}})$, is spanned by the element given in the right-hand side of (B.15).

By considering explicit expressions for $(z_2^2-1) {\mathcal {L}}_\varrho (z_1)_{13} {\mathcal {L}}^-_{{{\bar{\varrho }}}}(z_2^{-1})_{32}^{-1}$ and $(z_2^2-1) {\mathcal {L}}^-_{{{\bar{\varrho }}}}(z_2^{-1})_{32}^{-1} {\mathcal {L}}_\varrho (z_1)_{13}$, we obtain that (B.18) amounts to the system

$$\begin{aligned}{} & {} X \big ( q^{D_1 - D_2 -1} - a^\dag _1 a_2 q^{-D_1 +D_2 -2} z \big ) = \big ( q^{D_1 - D_2 -1} - a_1 {{\bar{a}}}^\dag _2 q^{D_1 - D_2} z \big ) X, \\{} & {} X \Big ( \big ( {{\bar{a}}}^\dag _2 q^{D_1 - D_2 -1} + a^\dag _1 q^{-D_1 - D_2 -2} z \big ) - a^\dag _1 q^{-D_1 + D_2} z z_2^2 \Big ) =\\{} & {} \quad = \Big ( {{\bar{a}}}^\dag _2 q^{-D_1-D_2-1} - \big ( a^\dag _1 q^{-D_1-D_2-2} + {{\bar{a}}}^\dag _2 q^{D_1-D_2+1} z \big ) z z_2^2 \Big ) X, \\{} & {} X \Big ( a_2 q^{-D_1+D_2-1} - \big ( a_1 q^{D_1-D_2} + a_2 q^{D_1+D_2+1} z \big ) z z_2^2 \Big ) = \\{} & {} \quad = \Big ( \big ( a_2 q^{D_1+D_2-1} + a_1 q^{D_1-D_2} z \big ) - a_1 q^{D_1+D_2+2} z z_2^2 \Big ) X, \\{} & {} X \Big ( q^{-D_1+D_2+1} + q^{D_1-D_2+1} z^2 - a_1 {{\bar{a}}}^\dag _2 q^{D_1-D_2} z \Big )\\{} & {} \quad = \Big ( q^{-D_1+D_2+1} + q^{D_1-D_2+1} z^2 - a^\dag _1 a_2 q^{-D_1+D_2-2} z \Big ) X \end{aligned}$$

for $X \in \widetilde{{\mathcal {A}}}_0^{(2)}$ fixed by $\chi ^{(2)}$. Since $\mathbb {C}[[z,z_2]] \cong (\mathbb {C}[[z]])[[z_2]]$, considering expansion coefficients with respect to $z_2$, we can use $[X,q^{D_1+D_2}]=0$ to deduce that the above system is equivalent to

$$\begin{aligned} X a_2 q^{-2D_1}&= {} \big ( a_2 + a_1 q^{-2D_2+1} z \big ) X,&\quad a_1 X&= X \big ( a_1 q^{-2(D_2+1)} + q^{-1} a_2 z \big ), \nonumber \\ X a^\dag _1 q^{2(D_2+1)}&= {} \big ( a^\dag _1 + {{\bar{a}}}^\dag _2 q^{2D_1+3} z \big ) X,\quad&{{\bar{a}}}^\dag _2 X&= X \big ( {{\bar{a}}}^\dag _2 q^{2D_1} + q^{-1} a^\dag _1 z \big ), \end{aligned}$$

(B.20)

$$\begin{aligned} \big [ X, q^{2D_1} \big ]=&{} \big ( X a^\dag _1 a_2 q^{2D_2-1} - a_1 {{\bar{a}}}^\dag _2 q^{2D_1+1} X \big ) z, \nonumber \\ \big [ X, q^{2D_2} + q^{2D_1} z^2 \big ]=&{} \big ( X a_1 {{\bar{a}}}^\dag _2 q^{2D_1-1} - a^\dag _1 a_2 q^{2D_2-3} X \big ) z. \end{aligned}$$

(B.21)

Note that $q^{-2D_1(D_2+1)} \in \widetilde{{\mathcal {A}}}^{(2)}_0$ is fixed by $\chi ^{(2)}$. Hence without loss of generality we may write

$$\begin{aligned} X = {\widetilde{X}} q^{-2D_1(D_2+1)}, \end{aligned}$$

(B.22)

for some ${\widetilde{X}} \in \widetilde{{\mathcal {A}}}^{(2)}_0$ fixed by $\chi ^{(2)}$. The system (B.20–B.21) is equivalent to

$$\begin{aligned}{}[{\widetilde{X}}, a_2]&= q^{-2D_2+1} a_1 {\widetilde{X}} z, \quad [a_1, {\widetilde{X}}] = {\widetilde{X}} q^{2D_1-1} a_2 z, \end{aligned}$$

(B.23)

$$\begin{aligned}{}[{\widetilde{X}}, a^\dag _1]&= {{\bar{a}}}^\dag _2 q^{2D_1+3} {\widetilde{X}} z, \quad [{{\bar{a}}}^\dag _2,{\widetilde{X}}] = {\widetilde{X}} a^\dag _1 q^{-2D_2-3} z, \end{aligned}$$

(B.24)

$$\begin{aligned} \big [ {\widetilde{X}}, q^{2D_1} \big ]&= \big ( {\widetilde{X}} a^\dag _1 a_2 q^{2D_1-1} - a_1 {{\bar{a}}}^\dag _2 q^{2D_1+1} {\widetilde{X}} \big ) z, \end{aligned}$$

(B.25)

$$\begin{aligned} \big [ {\widetilde{X}}, q^{2D_2} + q^{2D_1} z^2 \big ]&= \big ( {\widetilde{X}} a_1 {{\bar{a}}}^\dag _2 q^{2D_2+3} - a^\dag _1 a_2 q^{2D_2-3} \widetilde{X} \big ) z. \end{aligned}$$

(B.26)

Since $\chi ^{(2)}$ fixes ${\widetilde{X}}$, the equations in (B.23) and the equations in (B.24) are pairwise equivalent. At the same time, the system (B.23–B.24) implies (B.25) and (B.26). To show this, since $[\widetilde{X},q^{2D_1}] = [a^\dag _1 a_1,{\widetilde{X}}]$ from (B.23–B.24) we obtain

$$\begin{aligned}&[{\widetilde{X}},q^{2D_1}] + a_1 {{\bar{a}}}^\dag _2 q^{2D_1+1} {\widetilde{X}} z - {\widetilde{X}} a^\dag _1 a_2 q^{2D_1-1} z = \\&\qquad = a_1 {{\bar{a}}}^\dag _2 q^{2D_1+1} {\widetilde{X}} z - [{\widetilde{X}},a^\dag _2 ] a_1 + a^\dag _1 [a_1,{\widetilde{X}}] - {\widetilde{X}} a^\dag _1 a_2 q^{2D_1-1} z \\&\qquad = \big ( {{\bar{a}}}^\dag _2 q^{2D_1+3} [a_1, {\widetilde{X}}] - [{\widetilde{X}},a^\dag _1] a_2 q^{2D_1-1} \big ) z, \end{aligned}$$

which vanishes, thereby recovering (B.25). Applying $\chi ^{(2)}$ to (B.25) we obtain $[{\widetilde{X}},q^{-2D_2}] = \big ( {\widetilde{X}} a^\dag _1 a_2 q^{-2D_2-1} - a_1 {{\bar{a}}}^\dag _2 q^{-2D_2+1} {\widetilde{X}} \big ) z$. Left-and-right multiplying this by $q^{2D_2}$ and using (B.23–B.24) to rewrite the result we obtain

$$\begin{aligned}{}[{\widetilde{X}},q^{2D_2}] = \big ( {{\bar{a}}}^\dag _2 {\widetilde{X}} a_1 q^{2D_2+3} - q^{2D_2-1} a^\dag _1 {\widetilde{X}} a_2 \big ) z. \end{aligned}$$

(B.27)

Finally, using (B.27) and again (B.23–B.24), we derive that

$$\begin{aligned}&[{\widetilde{X}},q^{2D_2} + q^{2D_1} z^2] - {\widetilde{X}} a_1 {{\bar{a}}}^\dag _2 q^{2D_2+3} z + a^\dag _1 a_2 q^{2D_2-3} {\widetilde{X}} z = \\&\qquad = {{\bar{a}}}^\dag _2 {\widetilde{X}} a_1 q^{2D_2+3} z - q^{2D_2-1} a^\dag _1 {\widetilde{X}} a_2 z + [{\widetilde{X}},q^{2D_1}] z^2 + \\&\qquad \qquad - ({{\bar{a}}}^\dag _2 {\widetilde{X}} - {\widetilde{X}} a^\dag _1 q^{-2D_2-3} z) a_1 q^{2D_2+3} z + a^\dag _1 q^{2D_2-1} ({\widetilde{X}} a_2 - a_1 q^{-2D_2+1} {\widetilde{X}} z) z \\&\qquad = \big ( {\widetilde{X}} a^\dag _1 a_1 + a^\dag _1 a_1 {\widetilde{X}} + [{\widetilde{X}}, 1 - a^\dag _1 a_1] \big ) z^2 \end{aligned}$$

which vanishes, thereby proving (B.26) as well.

We have obtained that the system (B.23–B.26) is equivalent to the system (B.24). Writing $p=q^2$, without loss of generality we set

$$\begin{aligned} {\widetilde{X}} = Y e_p(q^3 z a_1 {{\bar{a}}}^\dag _2) e_p(q^{-1} z a^\dag _1 a_2) \end{aligned}$$

for some $Y \in \widetilde{{\mathcal {A}}}^{(2)}_0$ fixed by $\chi ^{(2)}$, noting that $e_p(q^3 z a_1 {{\bar{a}}}^\dag _2)$ and $e_p(q^{-1} z a^\dag _1 a_2)$ lie in $\widetilde{{\mathcal {A}}}^{(2)}_0$ and are fixed by $\chi ^{(2)}$. The theorem now follows from the following claim.

Claim: (B.24) is satisfied if and only if $Y \in \mathbb {C}[[z]]$.

In the special case $Y=1$, (B.24) is indeed satisfied:

$$\begin{aligned}&[{\widetilde{X}}, a^\dag _1] - {{\bar{a}}}^\dag _2 q^{2D_1+3} z {\widetilde{X}}= \\&\quad = \Big ( [ e_p(q^3 z a_1 {{\bar{a}}}^\dag _2), a^\dag _1] - {{\bar{a}}}^\dag _2 q^{2D_1+3} z e_p(q^3 z a_1 {{\bar{a}}}^\dag _2) \Big ) e_p(q^{-1} z a^\dag _1 a_2), \\ {[}&{{\bar{a}}}^\dag _2, {\widetilde{X}}] - {\widetilde{X}} a^\dag _1 q^{-2D_2-3} z=\\&\quad = e_p(q^3 z a_1 {{\bar{a}}}^\dag _2) \Big ( [{{\bar{a}}}^\dag _2, e_p(q^{-1} z a^\dag _1 a_2)] - e_p(q^{-1} z a^\dag _1 a_2) a^\dag _1 q^{-2D_2-3} z \Big ), \end{aligned}$$

with the expressions in parentheses vanishing by virtue of (A.14) and (B.10) (with $y=q^{-1}z$).

For general Y we therefore have

$$\begin{aligned}{}[{\widetilde{X}}, a^\dag _1] - {{\bar{a}}}^\dag _2 q^{2D_1+3} z {\widetilde{X}}&= [Y,a^\dag _1] e_p(q^3 z a_1 {{\bar{a}}}^\dag _2) e_p(q^{-1} z a^\dag _1 a_2), \\ {[}{{\bar{a}}}^\dag _2, {\widetilde{X}}] - {\widetilde{X}} a^\dag _1 q^{-2D_2-3} z&= [{{\bar{a}}}^\dag _2, Y] e_p(q^3 z a_1 {{\bar{a}}}^\dag _2) e_p(q^{-1} z a^\dag _1 a_2). \end{aligned}$$

Both right-hand sides vanish, i.e. (B.24) is indeed satisfied, if and only if Y lies in the centralizer in $\widetilde{{\mathcal {A}}}^{(2)}$ of $\{ a^\dag _1, {{\bar{a}}}^\dag _2 \}$, which is trivial by Lemma B.5. This proves the claim. $\square$

An Alternative Proof of the Main Theorem

In this part of the appendix we give a proof of the boundary factorization identity (8.2) independent of the universal K-matrix formalism, instead relying on the explict expressions obtained in Appendix B. Before we state and prove a key lemma, note that expressions of the form $e_p(\gamma ^D y)$ where $\gamma \in \mathbb {C}^\times$ and y is an indeterminate give rise to well-defined ${{\,\textrm{End}\,}}(W)$-valued formal power series, sending $w_j$ to $e_p(\gamma ^j y) w_j$.

Lemma C.1

Let y be a formal parameter and let p be a nonzero complex number, not a root of unity. In ${{\,\textrm{End}\,}}(W \otimes W)[[y]]$ we have the identities

$$\begin{aligned} e_p(p a_1 {{\bar{a}}}^\dag _2) (y;p)_{D_1}&= (y;p)_{D_1} e_p(-a_1 {{\bar{a}}}^\dag _2 p^{D_1} y) e_p(p a_1 {{\bar{a}}}^\dag _2) \end{aligned}$$

(C.1)

$$\begin{aligned} e_p(p a_1 {{\bar{a}}}^\dag _2) (p^{1-D_1} y;p)_{D_1}^{-1} e_p(p y {{\bar{a}}}^\dag _1 a_2)&= e_p(p y {{\bar{a}}}^\dag _1 a_2) (p^{1-D_2} y;p)_{D_2}^{-1} e_p(p a_1 {{\bar{a}}}^\dag _2). \end{aligned}$$

(C.2)

Proof

Note that

$$\begin{aligned} W \otimes W = \bigoplus _{m \in \mathbb {Z}_{\geqslant 0}} (W \otimes W)_m, \qquad (W \otimes W)_m:= \bigoplus _{j,k \geqslant 0 \atop j+k=m} \mathbb {C}w_j \otimes w_k. \end{aligned}$$

Because each factor in (C.1–C.2) preserves each finite-dimensional subspace $(W \otimes W)_m$, it suffices to prove the restrictions of (C.1–C.2) to $(W \otimes W)_m$, where $m \in \mathbb {Z}_{\geqslant 0}$ is fixed but arbitrary. Note that on $(W \otimes W)_m$ the operators appearing as arguments of the deformed exponentials are nilpotent. Therefore the operators on the left- and right-hand side of the restricted equations depend rationally on p and hence it suffices to prove them with p restricted to an open subset of $\mathbb {C}$.

We will prove the restriction of (C.1) to $(W \otimes W)_m$ for all $p \in \mathbb {C}$ such that $|p|<1$. Combining (A.4) and (A.7) we obtain $(y;p)_D = \frac{e_p(p^Dy)}{e_p(y)}$; as a consequence, (C.1) is equivalent to

$$\begin{aligned} e_p(p a_1 {{\bar{a}}}^\dag _2) e_p(p^{D_1} y) = e_p(p^{D_1} y) e_p(-a_1 {{\bar{a}}}^\dag _2 p^{D_1} y) e_p(p a_1 {{\bar{a}}}^\dag _2). \end{aligned}$$

(C.3)

But this equation follows directly from (A.12) and the observation $(a_1 {{\bar{a}}}^\dag _2)(p^{D_1}y) = p(p^{D_1}y)(a_1 {{\bar{a}}}^\dag _2)$.

On the other hand,¹³ we will prove the restricted version of (C.2) for all $p \in \mathbb {C}^\times$ such that $|p|>1$. In this case, for all $j \in \mathbb {Z}_{\geqslant 0}$ we have

$$\begin{aligned} (p^{1-j}y;p)_j^{-1} = (y;p^{-1})_j^{-1} = \frac{(p^{-j}y;p^{-1})_\infty }{(y;p^{-1})_\infty } \in \mathbb {C}[[y]]. \end{aligned}$$

From (A.7) and (A.9) we deduce the identity

$$\begin{aligned} (p^{-j}y;p^{-1})_\infty = e_{p^{-1}}(p^{-j} y)^{-1} = e_p(p^{1-j} y) \in \mathbb {C}[[y]]. \end{aligned}$$

Hence $(p^{1-D}y;p)_D^{-1} = (y;p^{-1})_\infty ^{-1} e_p(p^{1-D}y)$ in ${{\,\textrm{End}\,}}(W)[[y]]$ and (C.2) is equivalent to

$$\begin{aligned} e_p(p a_1 {{\bar{a}}}^\dag _2) e_p(p^{1-D_1}y) e_p(p y {{\bar{a}}}^\dag _1 a_2) = e_p(p y {{\bar{a}}}^\dag _1 a_2) e_p(p^{1-D_2}y) e_p(p a_1 {{\bar{a}}}^\dag _2). \end{aligned}$$

(C.4)

To prove (C.4), note that (B.11) can be evaluated at $y=p$, and the result can be rewritten as

$$\begin{aligned} e_p(p a_1 {{\bar{a}}}^\dag _2) \big ( p^{-D_1} + {{\bar{a}}}_1^\dag a_2 \big ) = \big ( p^{-D_2} + {{\bar{a}}}_1^\dag a_2 \big ) e_p(p a_1 {{\bar{a}}}^\dag _2). \end{aligned}$$

By iteration we obtain

$$\begin{aligned} e_p(p a_1 {{\bar{a}}}^\dag _2) e_p\big ( p^{1-D_1}y + p y {{\bar{a}}}^\dag _1 a_2 \big ) = e_p\big ( p^{1-D_2} y + p y {{\bar{a}}}^\dag _1 a_2 \big ) e_p(p a_1 {{\bar{a}}}^\dag _2). \end{aligned}$$

(C.5)

Note that $({{\bar{a}}}^\dag _1 a_2) p^{1-D_1} = p \, p^{1-D_1} ({{\bar{a}}}^\dag _1 a_2)$ and $p^{1-D_2} ({{\bar{a}}}^\dag _1 a_2) = p \, ({{\bar{a}}}^\dag _1 a_2) p^{1-D_2}$. Applying (A.11), we obtain (C.4), as required. $\square$

Alternative proof of Theorem 8.1

By virtue of (4.12), the desired identity, viz.

$$\begin{aligned} {\mathcal {K}}_\upsilon (z)_1 {\mathcal {R}}_{\upsilon \phi }(z^2) {\mathcal {K}}_\phi (z)_2 \,{\mathcal {O}}= {\mathcal {O}}{\mathcal {K}}_\varrho (q^{-\mu /2}z)_1 {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z^2) {\mathcal {K}}_{{\bar{\varrho }}}(q^{\mu /2}z)_2 \end{aligned}$$

(C.6)

for arbitrary $\mu \in \mathbb {C}$ and $q, \xi \in \mathbb {C}^\times$ such that q is not a root of unity, is equivalent to

$$\begin{aligned}{} & {} e_{p}(p a_1 {{\bar{a}}}^\dag _2) {\mathcal {K}}_\upsilon (z)_1 {\mathcal {R}}_{\upsilon \phi }(z^2) {\mathcal {K}}_\phi (z)_2 e_{p}(p a_1 {{\bar{a}}}^\dag _2)^{-1} = \nonumber \\{} & {} \qquad = q^{\mu (D_1-D_2)/2} {\mathcal {K}}_\varrho (q^{-\mu /2}z)_1 {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z^2) {\mathcal {K}}_{{\bar{\varrho }}}(q^{\mu /2}z)_2 q^{\mu (D_2-D_1)/2} \end{aligned}$$

(C.7)

where $p=q^2$. The strategy of the proof is as follows. We move various simple factors in ${\mathcal {F}}^{(2)}(D_1,D_2)$ to the right in both sides of (C.7), thus bringing them to a similar form. Then more advanced product formulas involving q-exponentials and finite q-Pochhammer symbols yield the desired equality.

More precisely, we set $\gamma = p q^{-\mu } \xi ^{-1} \in \mathbb {C}^\times$ and from (A.2) deduce

$$\begin{aligned} (\gamma z^{-2};p)_j^{-1} = p^{j(1-j)/2} (-\gamma ^{-1} z^2)^j (p^{1-j}\gamma ^{-1} z^2;p)_j^{-1} \end{aligned}$$

for all $j \in \mathbb {Z}_{\geqslant 0}$. Using the identities $q^{-D^2} a^\dag = -q {{\bar{a}}}^\dag q^{-D^2}$ and $q^{-D^2} a = a q^{2D-1} q^{-D^2}$, we obtain, for the left-hand side of (C.7),

$$\begin{aligned}&{} e_{p}(p a_1 {{\bar{a}}}^\dag _2) {\mathcal {K}}_\upsilon (z)_1 {\mathcal {R}}_{\upsilon \phi }(z^2) {\mathcal {K}}_\phi (z)_2 e_{p}(p a_1 {{\bar{a}}}^\dag _2)^{-1}= \nonumber \\{}&{} \; = e_{p}(p a_1 {{\bar{a}}}^\dag _2) \big ( -\gamma ^{-1} q^{1-D_1} \big )^{D_1} (\gamma z^2;p)_{D_1} (p^{1-D_1}\gamma ^{-1} z^2;p)_{D_1}^{-1} e_{p}( z^{2} a^\dag _1 a_2) \cdot \nonumber \\{}&{} \; \qquad \cdot q^{(2\mu -1)D_1-2D_2 - 2D_1D_2 - D_2^2} (-\xi )^{D_2} e_{p}(p a_1 {{\bar{a}}}^\dag _2)^{-1} \nonumber \\{}&{} \; = e_{p}(p a_1 {{\bar{a}}}^\dag _2) (\gamma z^2;p)_{D_1} (p^{1-D_1}\gamma ^{-1} z^2;p)_{D_1}^{-1} \cdot \nonumber \\{}&{} \; \qquad \cdot e_{p}(p \gamma ^{-1} z^2 {{\bar{a}}}^\dag _1 a_2) (-q^{-D_1-D_2-2} \xi )^{D_1+D_2} e_{p}(p a_1 {{\bar{a}}}^\dag _2)^{-1} \nonumber \\{}&{} \; = e_{p}(p a_1 {{\bar{a}}}^\dag _2) (\gamma z^2;p)_{D_1} (p^{1-D_1} \gamma ^{-1} z^2;p)_{D_1}^{-1} \cdot \nonumber \\{}&{} \; \qquad \cdot e_{p}(p \gamma ^{-1} z^2 {{\bar{a}}}^\dag _1 a_2) e_{p}(p a_1 {{\bar{a}}}^\dag _2)^{-1} (-q^{-D_1-D_2-2} \xi )^{D_1+D_2}. \end{aligned}$$

(C.8)

Similarly, for the right-hand side of (C.7) we obtain

$$\begin{aligned}{} & {} q^{\mu (D_1-D_2)/2} {\mathcal {K}}_\varrho (q^{-\mu /2}z)_1 {\mathcal {R}}_{\varrho {{\bar{\varrho }}}}(z^2) {\mathcal {K}}_{{\bar{\varrho }}}(q^{\mu /2}z)_2 q^{\mu (D_2-D_1)/2}= \nonumber \\{} & {} \; = (\gamma z^2;p)_{D_1} q^{\mu (D_1-D_2)/2 - D_1^2} (-\xi )^{D_1} e_{p}(q^3 z^2 a_1 {{\bar{a}}}^\dag _2) e_{p}(q^{-1} z^2 a_1^\dag a_2) \cdot \nonumber \\{} & {} \; \qquad \cdot (p^{1-D_2} \gamma ^{-1} z^2;p)_{D_2}^{-1} q^{\mu (D_2-D_1)/2 -2(D_1+D_2)-2D_1D_2-D_2^2} (-\xi )^{D_2} \nonumber \\{} & {} \; = (\gamma z^2;p)_{D_1} e_{p}(-a_1 {{\bar{a}}}^\dag _2 q^{2D_1} \gamma z^2)\nonumber \\{} & {} \; \qquad e_{p}(p \gamma ^{-1} z^2 {{\bar{a}}}^\dag _1 a_2) (p^{1-D_2} \gamma ^{-1} z^2;p)_{D_2}^{-1} (-q^{-D_1-D_2-2} \xi )^{D_1+D_2}. \end{aligned}$$

(C.9)

Therefore (C.7) is equivalent to

$$\begin{aligned}{} & {} e_{p}(p a_1 {{\bar{a}}}^\dag _2) (\gamma z^2;p)_{D_1} (p^{1-D_1} \gamma ^{-1} z^2;p)_{D_1}^{-1} e_{p}(p \gamma ^{-1} z^2 {{\bar{a}}}^\dag _1 a_2) e_{p}(p a_1 {{\bar{a}}}^\dag _2)^{-1} = \nonumber \\{} & {} \quad = (\gamma z^2;p)_{D_1} e_{p}(-a_1 {{\bar{a}}}^\dag _2 p^{D_1} \gamma z^2) e_{p}(p \gamma ^{-1} z^2 {{\bar{a}}}^\dag _1 a_2) (p^{1-D_2} \gamma ^{-1} z^2;p)_{D_2}^{-1}.\nonumber \\ \end{aligned}$$

(C.10)

Applying (C.1) with $y=\gamma z^2$ and (C.2) with $y = \gamma ^{-1} z^2$, we deduce (C.10), as required.    $\square$