Braid group action and root vectors for the $q$-Onsager algebra

We define two algebra automorphisms $T_0$ and $T_1$ of the $q$-Onsager algebra $B_c$, which provide an analog of G. Lusztig's braid group action for quantum groups. These automorphisms are used to define root vectors which give rise to a PBW basis for $B_c$. We show that the root vectors satisfy $q$-analogs of Onsager's original commutation relations. The paper is much inspired by I. Damiani's construction and investigation of root vectors for the quantized enveloping algebra of $\widehat{\mathfrak{sl}}_2$.


Introduction
The Onsager algebra O appeared first in 1944 in L. Onsager's investigation of the two-dimensional Ising model [Ons44]. It is an infinite dimensional Lie algebra with two natural presentations in terms of generators and relations. Onsager's original definition provides a linear basis {A n , G m | n ∈ Z, m ∈ N} and the commutators see [DG82]. With the advent of the general theory of Kac-Moody algebras [Kac90], it became clear that the Onsager algebra is isomorphic to the Lie subalgebra of the affine Lie algebra sl 2 consisting of all elements fixed under the Chevalley involution. The loop realization of sl 2 leads to the presentation (1.1) while the realization in terms of a Cartan matrix leads to the presentation (1.2). In particular, the two presentations define isomorphic Lie algebras, see also [Dav91,Roa91].
The q-Onsager algebra B c is a quantum group analog of the universal enveloping algebra U (O). In the present paper it depends on a parameter c ∈ Q(q) and q is a formal variable. Initially, the q-Onsager algebra was defined in terms of generators and q-analogs of the Dolan-Grady relations (1.2), see [Ter99], [Bas05]. The same relations showed up earlier in the context of polynomial association schemes [Ter93]. The q-Onsager algebra B c can be realized as a left or right coideal subalgebra of the quantized enveloping algebra U q ( sl 2 ), see [BB10], [BB12], [Kol14]. It is the simplest example of a quantum symmetric pair coideal subalgebra of affine type, see [Kol14,Example 7.6].
It is an open problem to find quantum group analogs of the generators A n , G m and of Onsager's relations (1.1). Attempts to find a current algebra realization of B c in the spirit of Drinfeld's second realization of U q ( sl 2 ) were made in [BK05], [BS10]. This was pursued further in [BB17] where it is conjectured that B c is isomorphic to a certain quotient of the current algebra A q proposed in [BS10]. The generators of the quotient of A q , however, do not specialize to the generators A n , G m of O.
In the present paper we construct quantum group analogs {B nδ+α1 , B mδ | n ∈ Z, m ∈ N} of Onsager's generators {A n , G m | n ∈ Z, m ∈ N} of O. We call these elements root vectors for B c . Let α 0 , α 1 denote the simple roots for sl 2 and set δ = α 0 + α 1 . Let R + = {nδ + α 0 , mδ, nδ + α 1 | n ∈ N 0 , m ∈ N } be the set of positive roots of sl 2 . For any n ∈ N define B (n−1)δ+α0 = B −nδ+α1 . Then we have generators B γ for any γ ∈ R + . Following [Dam93] we introduce an ordering on the set of positive roots R + . Using a filtered-graded argument, we prove a Poincaré-Birkhoff-Witt Theorem for the root vectors. We show that the root vectors B γ for γ ∈ R + satisfy commutation relations which are q-analogs of the Onsager relations (1.1). For any p ∈ Q(q) and x, y ∈ U q ( sl 2 ) define the p-commutator by [x, y] p = xy − pyx. Recall the notation [2] q = q + q −1 . The following theorem is the main result of this paper.
Theorem II. For any m, n ∈ N, p ∈ N 0 , r ∈ Z, and i ∈ {0, 1} the following relations hold where C re r,m and C im p,m,i are linear combinations of elements of the PBW-basis with coefficients in (q − 1) Z[q, q −1 ] + Z[q, q −1 ]c .
The commutation relations (1.3), (1.4), and (1.5) specialize to the Onsager relations (1.1) for q → 1, up to rescaling of the root vectors, see Section 3.3. The elements C re r,m and C im p,m,i are given explicitly in Propositions 5.5, 5.6, 5.9 and 5.10. The construction of the root vectors B γ for γ ∈ R + and the proof of Theorem II are much inspired by I. Damiani's construction and investigation of root vectors for U q ( sl 2 ) in [Dam93]. Damiani constructs root vectors for the positive part U + of U q ( sl 2 ). She uses G. Lusztig's braid group action of the free group in two generators on U q ( sl 2 ) to construct real root vectors {E nδ+α0 , E nδ+α1 | n ∈ N 0 } in U + . Subsequently she obtains imaginary roots vectors {E nδ | n ∈ N} as quadratic expressions in the real root vectors. She proves commutation formulas for the E β , β ∈ R + , by a subtle inductive procedure.
It was conjectured in [KP11, Conjecture 1.2] that quantum symmetric pairs of finite type have a natural action of a braid group, and it is reasonable to expect that such an action also exists in the Kac-Moody case. The Onsager algebra is invariant under the action of the braid group of sl 2 , which is the free group in two generators. Hence we expect to find two suitable algebra automorphisms of the q-Onsager algebra B c . We construct these automorphisms, T 0 and T 1 , in Section 2.3 and use them to define real root vectors {B nδ+α0 , B nδ+α1 | n ∈ N 0 } very much in the spirit of [Dam93]. The q-Onsager algebra B c is filtered with associated graded algebra U + . The imaginary root vectors {B mδ | m ∈ N} are again defined as quadratic expressions in the real root vectors, however these expressions now involve additional terms of lower filter degree, see Section 3.2. Once all root vectors are defined, the PBW basis of Theorem I is established by a filtered-graded argument using the PBW theorem for U + in [Dam93] and facts about the structure of quantum symmetric pairs, see [Kol14]. The commutation relations in Theorem II are again proved by an inductive calculation. This calculation is significantly harder than the corresponding calculations in [Dam93] due to the lower order terms in the definition of the imaginary root vectors B mδ .
The fact that the coefficients in Theorem II lie in (q − 1) Z[q, q −1 ] + Z[q, q −1 ]c suggests a connection to integral forms. In particular, it is natural to ask whether the results of the present paper still hold if we relax our assumptions about q. The answer is not straightforward and shall be considered elsewhere.
The paper is organized as follows. In Section 2 we recall the definition of the Onsager algebra O and the q-Onsager algebra B c and we introduce the braid group action on B c . The fact that T 0 and T 1 are indeed well-defined algebra automorphisms of B c is checked by computer calculations which are not reproduced here. In Section 3 we use the automorphisms T 0 and T 1 to define real and imaginary root vectors in B c . We show that up to a scalar factor these root vectors specialize to the Onsager generators A n , G m . In Section 4 we establish Theorem I. We first recall the PBW theorem for U + as proved in [Dam93]. We then show in Proposition 4.2 that up to a factor the root vectors B γ ∈ B c project onto Damiani's root vectors E γ ∈ U + in the associated graded algebra.
In Section 5, which forms the bulk of the paper, we prove the commutation relations of Theorem II explicitly. We first deal with the q −2 -commutator of two real root vectors in Propositions 5.5 and 5.6. The hardest part are the commutators of a real with an imaginary root vector which are established in Propositions 5.9 and 5.10. These commutators involve a crucial term F n for n ≥ 2. The term F n satisfies a recursive formula, the proof of which is deferred to Appendix A. Once the commutators [B nδ , B 1 ] are known, the fact that the imaginary root vectors commute is proved as in [Dam93].
Acknowledgements. The authors are grateful to István Heckenberger for checking the existence of the algebra automorphisms T 0 and T 1 with the computer algebra program FELIX in July 2013. We are much indebted to Travis Scrimshaw for pointing out an omission in the calculation of commutators of real root vectors in a previous version of this paper. His comments led to the formulation of Proposition 5.6 and to a reformulation of Equation (1.5) in the main Theorem II. The authors also wish to thank Marta Mazzocco and Paul Terwilliger for their interest in this work and for comments. PB visited the the School of Mathematics and Statistics at Newcastle University in July 2013 when this project started out. SK visited the Laboratoire de Mathématiques et Physique Théorique at Université Tours in May 2014 and April in 2017. Both authors are grateful to the hosting institutions for the hospitality and the good working conditions.
Note. When we were in the final stages of writing the present paper, Paul Terwilliger published the preprint [Ter17] which proves the existence of the algebra automorphisms T 0 and T 1 without the use of computer calculations.
Travis Scrimshaw has used the results of the present paper to implement the q-Onsager algebra in SageMath. His implementation is presently awaiting approval.

Braid group action on the q-Onsager algebra
In this introductory section we recall the definition of the Onsager algebra O and its q-analog B c . We then define the algebra automorphisms T 0 and T 1 of B c which are analogs of the Lusztig automorphisms for U q ( sl 2 ).
2.1. The Onsager algebra. Let e, f, h denote the standard generators of the Lie algebra sl 2 (C). The Chevalley involution θ : sl 2 (C) → sl 2 (C) is the involutive Lie algebra automorphism determined by We are interested in the affine Lie algebra where (·, ·) denotes the symmetric invariant bilinear form on sl 2 (C) with (e, f ) = 1, see [Kac90,Chapter 7]. The Chevalley involution θ : sl 2 → sl 2 is given by The Onsager algebra O is the infinite dimensional Lie subalgebra of sl 2 defined by The triangular decomposition of sl 2 implies that the following elements form a basis of O as a complex vector space Using the relations (2.1) one sees that A n and G m satisfy the relations (1.1). These relations first appeared in 1944 in Onsager's investigation of the Ising model [Ons44,(60), (61), (61a)]. Let α 0 , α 1 denote the simple roots of sl 2 and let δ = α 0 +α 1 denote the minimal positive imaginary root. We call A n the real root vector associated to the root nδ + α 1 if n ≥ 0, or associated to the root −nδ + α 1 = (−n + 1)δ + α 0 if n ≤ −1. Similarly, we call G m the imaginary root vector associated to the root mδ.
It follows from (1.1) that the Onsager algebra O is generated by the elements A 0 , A 1 as a Lie algebra. Defining relations can be seen to be the Dolan-Grady relations (1.2) which were discovered in [DG82]. Observe that A 1 is the real root vector associated to the root δ + α 1 . For our purposes it is more convenient to work with generators which are real root vectors associated to the simple roots α 0 , α 1 . To this end define The elements D 0 and D 1 also generate the Lie algebra O. The defining relations are now given by 2.2. The q-Onsager algebra B c . Let Q(q) denote the field of rational functions in an indeterminate q and let c ∈ Q(q) such that c(1) = 1. The q-Onsager algebra B c is the unital Q(q)-algebra generated by two elements B 0 , B 1 with the defining relations where we use the usual q-binomial coefficients given by At the specialization q → 1 the relations (2.4) transform into the modified Dolan-Grady relations (2.3). The q-Onsager algebra is the simplest example of a quantum symmetric pair coideal subalgebra for an affine Kac-Moody algebra. Indeed, let U q ( sl 2 ) denote the Drinfeld-Jimbo quantized enveloping algebra of the affine Kac-Moody algebra sl 2 with standard generators E 0 , E 1 , F 0 , F 1 , K ±1 0 , K ±1 1 . Then there exists an algebra embedding such that ι(B c ) is a right coideal subalgebra of U q ( sl 2 ), see [Kol14,Example 7.6].
To match the conventions in [Dam93] it is preferable to work with the algebra embedding is a left coideal subalgebra of U q ( sl 2 ), see also [BB12,(3.15)].
Remark 2.1. In [Kol14, Example 7.6] the q-Onsager algebra depends on 2 parameters c 0 , c 1 . Over a field which contains all square roots these algebras are isomorphic for any parameters. Here we choose to retain one parameter c = c 0 = c 1 because occasionally different choices for c are convenient. However, we do not keep two parameters c 0 , c 1 because this would complicate the definition of the algebra automorphism Φ in the upcoming Section 2.3.

Automorphisms of
The other two automorphisms are obtained from what appears to be a general principle, namely that quantum symmetric pair coideal subalgebras come with an action of a braid group [KP11]. In the case of B c one expects an action of the braid group of type A (1) 1 which is the free group in two generators. This action is closely related to Lusztig's braid group action on U q ( sl 2 ) but it is not merely obtained by restriction to the coideal subalgebra.
The inverse automorphism is given by This proposition was checked for us by István Heckenberger using the computer algebra program FELIX [AK]. He verified that T 0 (B 0 ) and T 0 (B 1 ) satisfy the defining relations (2.4) of B c , and similarly for T −1 0 (B 0 ) and T −1 0 (B 1 ), and he confirmed that T 0 (T −1 0 (B 1 )) = B 1 = T −1 0 (T 0 (B 1 )). The second braid group automorphism T 1 : B c → B c is defined by In words, T 1 is obtained from T 0 by exchanging subscripts 0 and 1 everywhere in (2.6) and (2.7).

The root vectors
In this section we define quantum analogs of the root vectors A n , G m of the Onsager algebra O up to scalar multiplication. Quantum analogs of A n will be called real root vectors, and quantum analogs of G m will be called imaginary root vectors of B c . To mimic Damiani's construction, the quantum analog of G m will be denoted by B mδ . Similarly, the quantum analog of A n will be denoted B nδ+α1 for n ≥ 0 and by B −(n+1)δ+α0 for n < 0, see Section 3.3 for the precise correspondence.
3.1. Definition of B δ and the real root vectors. Following Damiani's construction [Dam93, Section 3.1] we set where a, b ∈ Q(q) are parameters which are still to be determined. For any elements x, y ∈ B c we write [x, y] = xy − yx to denote their commutator. One calculates Here we have a choice. Indeed, up to an overall factor we can consider either Observe that Φ(B δ ) =B δ . As in Damiani's construction the choice of B δ dictates the choice of roots vectors B nδ+αi with i = 0, 1. Indeed, Equation (3.1) suggests to define and We then obtain We can now check an analog of [Dam93, Section 3.2, Lemma 1].
Lemma 3.1. The following relations hold in B c . ( Proof. To verify (1) we calculate To verify the first formula in (2) we apply . Using the result of (1) and the relation Φ(B δ ) =B δ one obtains the desired formula. The second formula in (2) is obtained analogously by applying so that the relations in part (2) of the above lemma become Remark 3.2. Formulas (3.6) and (3.7) are only valid for n ≥ 1. For n = 0 one obtains (3.4) and (3.5), respectively. This suggests to work with the following convention: for Remark 3.3. In a similar way, we could have definedB nδ+α0 = (T −1 0 Φ) n (B 0 ) and B nδ+α1 = (T 1 Φ) n (B 1 ).

3.2.
Definition of the imaginary root vectors B mδ for m ≥ 2. In view of the Onsager relations (1.1) we aim to construct commuting elements B mδ for m ≥ 1. Moreover, in view of Damiani's construction these elements should be fixed by T 0 Φ and they should be of the form where l.o.t denotes terms of lower degree in the generators B 0 , B 1 of B c . As we will show, the following definition will give the desired properties Observe in particular that C 1 = 0 and C 2 = B 2 1 . 3.3. Specialization of root vectors. It is well known that the q-Onsager algebra B c specializes to the Onsager algebra for q → 1, see e.g. [Kol14, Theorem 10.8]. For any element x ∈ B c we write x ∈ O to denote its specialization if it exists. See [Kol14, Section 10] for the precise notion of specialization used. By (2.2) and (2.5) one has We now compare the root vectors defined in the previous section to the root vectors defined by Damiani in [Dam93] for the positive part of U q ( sl 2 ). Using a filtered-graded argument this will allow us to prove a PBW theorem for B c in terms of the specific root vectors defined in Section 3.
consists of the linear span of all monomials of degree at most n in the generators Let Gr F (B c ) denote the associated graded algebra. Similarly, there exists a filtration G * of U + such that G n U + consists of the linear span of all monomials of degree at most n in the generators E 0 , E 1 of U + . It follows from [Kol14, Proposition 6.2] that The defining relations (2.4) of B c imply that the associated graded algebra Gr F (B c ) is a quotient of the graded algebra U + . This fact, together with relation (4.2) implies that For any n ∈ N 0 let π n : F n B c → F n B c /F n−1 B c denote the natural projection map. Using (4.3) we consider the image of the map π n as a subset of U + .
If γ = mδ is an imaginary root then Properties (1) and (2) for real roots and the definition (3.8) imply that π nγ (B mδ ) ∈ F nγ B c . Moreover, comparing (3.8) with (4.1) one obtains This completes the proof of the proposition.
As a consequence of the above proposition we immediately obtain the desired PBW theorem for B c . Recall the ordering < on R + defined in Subsection 4.1.
Theorem 4.3. The set of ordered monomials Proof. For n ∈ N consider the set of monomials Proposition 4.2.
(1) implies that B n ⊆ F n B c . By Proposition 4.2.
(2) and the PBW Theorem 4.1 for U + the elements of B n are linearly independent. Moreover, again by Theorem 4.1, the set B n contains dim Q(q) (G n U + ) = dim Q(q) (F n B c ) many elements. Hence B n is a basis of F n B c and the theorem follows.

Commutation relations
We now turn to the study of commutation relations inside B c . This provides q-analogs of the classical Onsager relations (1.1). Again one can mimic Damiani's approach to establish commutation relations for the root vectors B γ for γ ∈ R + . 5.1. Imaginary root vectors and braid group action. Recall the definition of B nδ and C n from Section 3.2. For n ≥ 2 one has (5.1) Equation (5.1) will play a crucial role in the proof of the following lemma. Moreover, for a fixed n ∈ N we repeatedly make the following assumption for all k < n. (A n I) Lemma 5.1. Let n ∈ N and assume that (A n I) holds. Then Proof. We use Equations (3.7), (5.1) to calculate Using (T 0 Φ) −1 (B (n−1)δ ) = B (n−1)δ which holds by (A n I) one obtains Inserting the above relation into (5.3) and using again (id − T 0 Φ)B (n−1)δ = 0 one gets Finally, we add up Equations (5.2) and (5.4) and obtain which completes the proof of the lemma.
Lemma 5.1 shows in particular that if (A k I) holds for some k ∈ N and additionally [B δ , B kδ ] = 0 then (A k+1 I) also holds. This observation has the following consequence.
Corollary 5.2. Let n ∈ N and assume that [B δ , B kδ ] = 0 for all k < n. Then (A n I) holds and Lemma 5.1 also allows us to rewrite the term C m given in (3.9) in ordered form with respect to the ordering < of R + defined in Subsection 4.1. For any real number x ∈ R we write [x] to denote the largest integer less than or equal to x.
Proposition 5.3. Let n ∈ N and assume that [B δ , B kδ ] = 0 for all k < n. Then for all m ∈ N with m ≤ n + 2 the relation holds, where the coefficients a m p are given by if m is even and p = m 2 . (5.6) Proof. As observed in Subsection 3.2, Equation (5.5) holds for m = 1, 2. Assume now that (5.5) holds for a given m ≤ n. By Corollary 5.2 one obtains (T 0 Φ) −1 (B mδ ) = B mδ . Hence On the other hand (5.1) implies that Adding the above two relations one obtains Using the induction hypothesis for C m this becomes where b m+2 Similarly to the recursive formula (5.1) for C n one has This is the desired second expression of B mδ . The element D m can be written in ordered form. The proof of the following proposition is identical to the proof of Proposition 5.3 and hence omitted.
Proposition 5.4. Let n ∈ N and assume that [B δ , B kδ ] = 0 for all k < n. Then for all m ∈ N with m ≤ n + 2 the relation holds, where the coefficients a m p are given by (5.6). 5.3. Commutators of real root vectors. For r, s ∈ N 0 the products B rδ+α1 B sδ+α1 and B sδ+α0 B rδ+α0 are ordered with respect to the ordering < of R + if and only if r ≥ s. We can use the definition of B mδ in (3.8) together with Proposition 5.3 to rewrite B rδ+α1 B sδ+α1 for r < s in ordered form. Similarly, we can use the second expression of B mδ in (5.8) together with Proposition 5.4 to rewrite B sδ+α0 B rδ+α0 for r < s in ordered form. For any p ∈ Q(q) and any x, y ∈ B c we write [x, y] p = xy − p yx to denote the p-commutator of x and y.
Proposition 5.5. Let n ∈ N and assume that [B δ , B kδ ] = 0 for all k < n. For all m ∈ N with m ≤ n and all r ∈ N 0 one has where the coefficients a m p are given by (5.6).
Next we rewrite the product B sδ+α1 B rδ+α0 for s, r ∈ N 0 as a sum of ordered products of root vectors.

5.4.
Commutators involving B nδ . We now describe the commutators of imaginary root vectors B nδ with any other root vector B γ for γ ∈ R + . For any n ∈ N with n ≥ 2 define As we will see, the elements F n play a crucial role in the description of the commutators [B nδ , B 0 ] and [B 1 , B nδ ]. The following recursive formula for F n will be proved in Appendix A.
Lemma 5.7. Let n ∈ N with n ≥ 2 and assume that [B δ , B kδ ] = 0 for all k < n. Then one has Straightforward calculations for m ≤ 4 suggest that the relations (5.16) and (5.17) below for the commutators B mδ , B 0 ] and B 1 , B mδ ] hold for every m ∈ N.
Lemma 5.8. Let n ∈ N and assume that [B δ , B kδ ] = 0 for all k < n. Then for all m ∈ N with m ≤ n the relations Replacing n by n − 1, we know in particular that In view of our assumption n ≥ 3 we can use Lemma 5.7 to obtain Replacing the above expression in (5.18) and simplifying, we obtain the recursive formula The three commutators on the right hand side of the above equation are known by induction hypothesis. One finds Inserting the three commutators above into (5.20) one obtains an equation which simplifies to Equation (5.16) for m = n.
As a consequence of Lemma 5.8 we can now describe the commutator of B mδ with any real root vector.
Proposition 5.9. Let n ∈ N and assume that [B δ , B kδ ] = 0 for all k < n. Then for all m ∈ N, p ∈ N 0 with m ≤ n and p ≤ m − 1 the following relations hold: Sketch of proof. We outline the calculations needed to show (5.21). For p = 0, Equation (5.21) holds by Lemma 5.8. We proceed by induction on p. Assume that (5.21) holds for some p with p + 2 ≤ m and consider (T 0 Φ) −1 B pδ+α1 , B mδ ] . The resulting expression is ordered except the term (T 0 Φ) −1 B 0 B (m−p−1)δ = B 1 B (m−p−1)δ which appears in the last sum. Using Formula (5.17) for m − p − 1, this term is rewritten as a linear combination of ordered monomials. Combining the resulting expressions one obtains (5.21) for p + 1. Equation (5.22) is verified analogously by application of (T 0 Φ).
Further application of (T 0 Φ) −1 and T 0 Φ extends the relations (5.22) and (5.21) to the case p ≥ m.
Proposition 5.10. Let n ∈ N and assume that [B δ , B kδ ] = 0 for all k < n. Then for all m, p ∈ N with m ≤ n and p ≥ m the following relations hold: Proof. First we show (5.23). One starts with (5.21) which, as just shown, holds for p = m − 1. Observe that the terms in the last two sums of (5.21) disappear for p = m − 1. Then one computes (T 0 Φ) −(p−m+1) [B (m−1)δ+α1 , B mδ ] which gives formula (5.23). Relation (5.24) is shown analogously.
Finally, we want to show that the commutators [B nδ , B mδ ] vanish. This will be achieved by an induction over the set of ordered pairs of natural numbers N 2 > = {(n, m) ∈ N × N | n > m} with the lexicographic ordering given by (k, l) < lex (n, m) ⇐⇒ k < n or (k = n and l < m). (5.25) As in [Dam93] it is convenient to first use the formula from Lemma 5.8 to show that the commutators [[B nδ , B mδ ], B 1 ] vanish. For the proof of the next lemma recall that by Remark 3.2 for k < 0 we write B kδ+α0 = B (−k−1)δ+α1 and B kδ+α1 = B (−k−1)δ+α0 .
Lemma 5.11. Let (n, m) ∈ N 2 > and assume that [B kδ , B lδ ] = 0 for all (k, l) < lex (n, m). Then Proof. Observe first that by Corollary 5.2 we know that for all k ≤ n.  Proof. Without loss of generality we may assume that m < n. We perform induction over the set of ordered pairs N 2 > with the lexicographic ordering given by (5.25). Assume that [B kδ , B lδ ] = 0 for all (k, l) ∈ N 2 > with (k, l) < lex (n, m). By Lemma 5.11 this implies that [B mδ , B nδ ] belongs to the center of B c . By [Kol14,Theorem 8.3] we know that the center of B c consists of scalars Q(q)1. However, [B mδ , B nδ ] can be written as a noncommutative polynomial in the generators B 0 , B 1 without a constant term. Such a polynomial can never be transformed into a scalar using only the q-Dolan Grady relations (2.4), unless the polynomial vanishes in B c .
Remark 5.13. The existence of a commutative polynomial ring in infinitely many variables inside B c initially came as a surprise to us. In the work on this paper, a crucial step towards establishing Proposition 5.12 was a brute force calculation showing that [B δ , B 2δ ] = 0. To this end we made an ansatz writing [B δ , B 2δ ] as an element in the ideal generated by the q-Dolan Grady expressions. This led to a system of equations which we could explicitly solve, showing that [B δ , B 2δ ] does indeed lie in the defining ideal of B c . While initially helpful, this method is too cumbersome to obtain Proposition 5.12 in full generality.
The above proposition together with Corollary 5.2 imply that the imaginary root vectors are fixed by T 0 Φ.
(3) Relation (5.13) holds for all r, s ∈ N 0 with r ≥ s. The relations in Proposition 5.12 and Corollary 5.15 provide the desired q-analogs of the Onsager relations (1.1). The above corollary implies Theorem II in the introduction. The explicit form of the terms C re r,m and C im p,m,i can be read off from Propositions 5.5, 5.6, 5.9 and 5.10.
Appendix A. Proof of Lemma 5.7 For any n ∈ N define The element R n will appear in the proof of Lemma 5.7. Observe that R 1 = R 2 = 0 and that Using the relation B δ = q −2 B 1 B 0 − B 0 B 1 one can rewrite the above expression as This formula has a generalization for all n ∈ N.
Lemma A.1. For any n ∈ N one has Proof. By (A.2) and the preceding comment we know already that Equation (A.3) holds for n = 1, 2, 3. Hence, for the remainder of this proof, assume that n ≥ 4. For any p ∈ N with 2 ≤ p ≤ n − 1 we introduce the notation With this notation we can write Similarly, for p ≥ 3 relation (5.1) implies that Again by (3.8) we have Using this formula in the first line of (A.7), one obtains which holds for p ≥ 3. Using the above expression one obtains for 3 ≤ p ≤ (n+ 1)/2 the relation (n, n−p) + (n, p−2) = (n, n−p) + (n, n−(n−p+2)) The terms (A.9) and (A.6) cover all terms of the sum (A.5) if n is odd. If n is even then the sum (A.5) contains the additional summand (n, n − ( n 2 + 1)). By (A.8) this summand is given by (n, n−( n 2 +1)) = − B ( n 2 −1)δ B ( n 2 −2)δ+α1 − q 2 B ( n 2 −2)δ+α1 B ( n 2 −1)δ (A.10) Adding up (A.6), (A.9) for 3 ≤ p ≤ (n + 1)/2, and (A.10) if n is even, we obtain As indicated, the second and the third line of the above expression vanish. Hence, to prove the lemma, it remains to see that the last line of the above expression also vanishes. This follows from the relation Replacing n by n − 3 and shifting the summation index up by 3, one obtains indeed that the last line of the above expression for R n vanishes. This completes the proof of the Lemma.
Hence we get the following formula which holds for all n ∈ N. Recall that by definition for any n ∈ N with n ≥ 2. Equation (A.12) describes the second commutator in the above expression. With the help of Lemma A.1 and Equation (A.12) we now provide an alternative formula for the element F n . This formula is the main ingredient needed to prove the recursive formula in Lemma 5.7.
Lemma A.2. Let n ∈ N with n ≥ 2 and assume that T 0 Φ(B kδ ) = B kδ for all k ∈ N with k ≤ n − 1. Then one has  We are now in a position to prove Lemma 5.7 as an immediate consequence of Lemma A.2.
Proof of Lemma 5.7. In view of the assumption [B δ , B kδ ] = 0 for all k < n, Corollary 5.2 implies that T 0 Φ(B kδ ) = B kδ for all k < n + 1. Hence Lemma A.2 implies that This proves Lemma 5.7.