Shifted Quantum Affine Algebras: Integral Forms in Type A

Abstract

We define an integral form of shifted quantum affine algebras of type A and construct Poincaré–Birkhoff–Witt–Drinfeld bases for them. When the shift is trivial, our integral form coincides with the RTT integral form. We prove that these integral forms are closed with respect to the coproduct and shift homomorphisms. We prove that the homomorphism from our integral form to the corresponding quantized K-theoretic Coulomb branch of a quiver gauge theory is always surjective. In one particular case we identify this Coulomb branch with the extended quantum universal enveloping algebra of type A. Finally, we obtain the rational (homological) analogues of the above results [proved earlier in Kamnitzer et al. (Proc Am Math Soc 146(2):861–874, 2018a; On category $\mathcal {O}$ for affine Grassmannian slices and categorified tensor products. arXiv:1806.07519, 2018b) via different techniques].

Appendices

Appendices By Alexander Tsymbaliuk and AlexWeekes.

1.1 Appendix A: PBW Theorem and Rees Algebra Realization for the Drinfeld–Gavarini Dual, and the Shifted Yangian

In Kamnitzer et al. (2014), dominantly shifted Yangians were defined for any semisimple Lie algebra ${\mathfrak {g}}$, generalizing Brundan–Kleshchev’s definition Brundan and Kleshchev (2006) for $\mathfrak {gl}_n$. Two issues with the definition given in Kamnitzer et al. (2014) are now clear:

• (a) Kamnitzer et al. (2014, §3C) recalled Drinfeld–Gavarini duality, and an explicit description of the Drinfeld–Gavarini dual based on the discussion in Gavarini (2002, §3.5). However, additional assumptions seem necessary in order for this description to be correct.

• (b) Applying the explicit description of (a), a presentation of the dominantly shifted Yangians was given in Kamnitzer et al. (2014, Theorem 3.5, Definition 3.10). But it was incomplete, as it does not include a full set of relations. In fact, writing down a complete (explicit) set of relations seems very difficult (at least, in terms of new Drinfeld generators).

We rectify (a) in Proposition A.2, which is of independent interest. We then verify that this result applies to the Yangian, yielding Theorem A.7. This proves that the set of generators given just before Kamnitzer et al. (2014, Theorem 3.5) is indeed correct. We also verify that Proposition A.2 applies to the RTT Yangian $Y^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)$, which implies an identification of its Drinfeld–Gavarini dual with the subalgebra ${\mathbf {Y}}^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)$ of Definition 2.3, establishes the PBW theorem for the latter (that we referred to in Sect. 2), and provides a conceptual proof of Proposition 2.21.

Another definition of the shifted Yangian (for an arbitrary, not necessarily dominant, shift) as a Rees algebra was given in Finkelberg et al. (2018, §5.4) and Braverman et al. (2016, Appendix B(i)), precisely to avoid the issues mentioned above. This raises

Question: Are these two definitions of the shifted Yangians equivalent for dominant shifts?

We answer this question in the affirmative in Theorem A.12, which generalizes Theorem 2.31 for any semisimple Lie algebra ${\mathfrak {g}}$.

We conclude this appendix with one more equivalent definition of the shifted Yangians, see Appendix A.8, in particular, Theorem A.17.

1.2 Drinfeld’s Functor

Let $\mathfrak {a}$ be a Lie algebra over ${\mathbb {C}}$. Assume that A is a deformation quantization of the Hopf algebra $U(\mathfrak {a})$,⁷ over ${\mathbb {C}}[\hbar ]$. In other words, A is a Hopf algebra over ${\mathbb {C}}[\hbar ]$, and there is an isomorphism of Hopf algebras $A/\hbar A \simeq U(\mathfrak {a})$.

Denote the coproduct and the counit of A by $\Delta$ and $\epsilon$, respectively. For any $n\ge 0$, let $\Delta ^n:A\rightarrow A^{\otimes n}$ be the n-th iterated coproduct (tensor product over ${\mathbb {C}}[\hbar ]$). It is defined inductively by $\Delta ^0=\epsilon$, $\Delta ^1={\text {id}}$, and $\Delta ^n=(\Delta \otimes {\text {id}}^{\otimes (n-2)} )\circ \Delta ^{n-1}$. Define $\delta _n:A\rightarrow A^{\otimes n}$ via

$$\begin{aligned} \delta _n:=({\text {id}} - \epsilon )^{\otimes n}\circ \Delta ^n. \end{aligned}$$

(A.1)

Drinfeld (1986) introduced functors on Hopf algebras, which have been studied extensively in work of Gavarini, see e.g. Gavarini (2002). In particular, the Drinfeld–Gavarini dual of A is the sub-Hopf algebra $A'\subset A$ defined by

$$\begin{aligned} A'=\left\{ a\in A : \delta _n(a)\in \hbar ^n A^{\otimes n} \ \mathrm {for\ all}\ n\in {\mathbb {N}}\right\} . \end{aligned}$$

(A.2)

As in the second part of the proof of Gavarini (2002, Proposition 2.6):

Lemma A.1

For any $a,b\in A'$, we have $[a,b] \in \hbar A'$.

The dual $A'$ can be defined for any Hopf algebra A over ${\mathbb {C}}[\hbar ]$. However, the case of the greatest interest is precisely when $A/\hbar A\simeq U(\mathfrak {a})$. In this case, one can prove that $A'$ is a deformation quantization of the coordinate ring of a ‘dual’ algebraic group. This is a part of the quantum duality principle, see Gavarini (2002, Theorem 1.6) (called Drinfeld–Gavarini duality in Kamnitzer et al. (2014, §3C)).

1.3 PBW Basis for $A'$

We will now make some additional assumptions on A. Suppose that there exists a totally ordered set $(\mathcal {I}, \le )$, and elements $\{ x_i \}_{i\in \mathcal {I}} \subset A$. By an (ordered) PBW monomial, we mean any ordered monomial

$$\begin{aligned} x_{i_1}\dots x_{i_\ell } \in A \end{aligned}$$

(A.3)

with $\ell \in {\mathbb {N}}$ and $i_1\le \ldots \le i_\ell$. Assume that:

We will use the multi-index notation $x^\alpha$ to denote a PBW monomial $\prod _{i\in \mathcal {I}} x_i^{\alpha _i}$ in the PBW generators $x_i$. We write $|\alpha |=\sum _{i\in \mathcal {I}} \alpha _i$ for the total degree of $x^\alpha$. Finally, for $\overline{a}\in U(\mathfrak {a})$ we denote by $\partial (\overline{a})$ its degree with respect to the usual filtration, i.e. the maximal value of $|\alpha |$ over all summands $\overline{x}^\alpha$ that appear in $\overline{a}$.

In Gavarini (2002, §3.5), an explicit description of $A'$ is given in the formal case, i.e. when working with complete algebras over ${\mathbb {C}}[[\hbar ]]$. The next result is inspired by this description, but with the aim of working instead over ${\mathbb {C}}[\hbar ]$.

Proposition A.2

Suppose that A satisfies Assumptions (As1)–(As3). Then $A'$ is free over ${\mathbb {C}}[\hbar ]$, with a basis given by the PBW monomials in the elements $\{\hbar x_i\}_{i\in \mathcal {I}}$. In particular, $A'\subset A$ is the ${\mathbb {C}}[\hbar ]$-subalgebra generated by $\{ \hbar x_i \}_{i\in \mathcal {I}}$.

In the proof, we will make use of Etingof and Kazhdan (1996, Lemma 4.12) (cf. (Gavarini 2002, Lemma 3.3)):

Lemma A.3

Let $a\in A'$ be non-zero, and write $a=\hbar ^n b$ where $b \in A{\setminus } \hbar A$. Then $\partial (\overline{b} )\le n$.

Proof of Proposition A.2

Let $c\in A'$. By assumption (As2), we can write $c=\sum _{k,\alpha } c_{k,\alpha }\hbar ^k x^\alpha$ for some $c_{k,\alpha }\in {\mathbb {C}}$ which are almost all zero. Since $A'$ is an algebra, by assumption (As3) we know that $\hbar ^k x^\alpha \in A'$ whenever $k\ge |\alpha |$. Subtracting all such elements from c, we conclude that the element

$$\begin{aligned} a=\sum _{k,\alpha :k < |\alpha |} c_{k,\alpha } \hbar ^k x^\alpha \ \in \ A'. \end{aligned}$$

(A.4)

Assume that $a\ne 0$. Choosing

$$\begin{aligned} n=\min \{k : \exists \alpha \ \mathrm {such\ that}\ k< |\alpha | \text { and } c_{k,\alpha } \ne 0\}, \end{aligned}$$

(A.5)

we can write $a = \hbar ^{n} b$, where

$$\begin{aligned} b\in \sum _{\alpha :n<|\alpha |} c_{n, \alpha }x^\alpha + \hbar A. \end{aligned}$$

(A.6)

From assumption (As1) it follows that

$$\begin{aligned} \overline{b}=\sum _{\alpha :n<|\alpha |} c_{n,\alpha } \overline{x}^\alpha \ \in \ U(\mathfrak {a}), \end{aligned}$$

(A.7)

and so $\partial (\overline{b})>n$. But by Lemma A.3 we should have $\partial (\overline{b})\le n$. So we conclude that $a=0$.

This shows that the PBW monomials in $\{\hbar x_i\}_{i\in \mathcal {I}}$ span $A'$ over ${\mathbb {C}}[\hbar ]$. But they are also linearly independent, because of assumption (As2). Thus, they form a basis.

1.4 Rees Algebra Description of $A'$

In this subsection, we make a further assumption on A:

Note that $A'\subset A$ is then a graded sub-Hopf algebra, and that the specializations of $A, A'$ at $\hbar =0$ inherit gradings. By Proposition A.2, we see that the inclusion $A'\subset A$ induces an isomorphism of their specializations at $\hbar =1$:

$$\begin{aligned} A'/(\hbar -1)A'\simeq A/(\hbar -1)A. \end{aligned}$$

(A.8)

Moreover, the images of their respective PBW generators and bases agree as $\hbar x_i + (\hbar -1)A =x_i + (\hbar -1) A$.

Denote the algebra in (A.8) by $A_{\hbar =1}$. If assumption (As4) holds, it follows that $A_{\hbar =1}$ inherits two filtrations $F'_\bullet A_{\hbar =1}, F_\bullet A_{\hbar =1}$, coming from $A'$ and A, respectively. Denoting $\mathrm {d}_i=\deg x_i$, these filtrations may be defined explicitly in terms of the PBW monomials:

$$\begin{aligned} F_k A_{\hbar =1}&= {\text {span}}_{{\mathbb {C}}} \Big \{ x^\alpha + (\hbar -1) A\ :\ \sum _i \mathrm {d}_i \alpha _i \le k \Big \}, \end{aligned}$$

(A.9)

$$\begin{aligned} F'_k A_{\hbar =1}&= {\text {span}}_{{\mathbb {C}}} \Big \{ x^\alpha +(\hbar -1) A\ :\ \sum _i (\mathrm {d}_i+1) \alpha _i \le k \Big \}. \end{aligned}$$

(A.10)

In particular, $F'_k A_{\hbar =1}\subset F_k A_{\hbar =1}$ for all $k\in {\mathbb {Z}}$.

By the above discussion, we obtain another description of $A'$, as a Rees algebra:

Proposition A.4

Suppose that A satisfies Assumptions (As1)–(As4). Then there is a canonical isomorphism of graded Hopf algebras

$$\begin{aligned} A'\simeq {\text {Rees}}^{F'_\bullet } (A_{\hbar =1}). \end{aligned}$$

It is compatible with the canonical isomorphism $A\simeq {\text {Rees}}^{F_\bullet }(A_{\hbar =1})$, under the natural inclusions $A'\subset A$ and ${\text {Rees}}^{F'_\bullet }(A_{\hbar =1})\subset {\text {Rees}}^{F_\bullet }(A_{\hbar =1})$.

1.5 The Yangian of ${\mathfrak {g}}$

Consider the Yangian$Y_\hbar =Y_\hbar ({\mathfrak {g}})$ associated to a semisimple Lie algebra ${\mathfrak {g}}$. It is the associative ${\mathbb {C}}[\hbar ]$-algebra with generators $\{e_i^{(r)}, h_i^{(r)}, f_i^{(r)}\}_{i\in I}^{r\in {\mathbb {N}}}$ (here I denotes the set of vertices of the Dynkin diagram of ${\mathfrak {g}}$), and relations as in Kamnitzer et al. (2014, §3A) and Guay et al. (2018, Definition 2.1), cf. (2.9). For each $i\in I$, define the element $s_i:=h_i^{(1)}-\tfrac{\hbar }{2} (h_i^{(0)})^2\in Y_\hbar$, cf. (2.17). Then $Y_\hbar$ is also generated by $\{e_i^{(0)}, h_i^{(0)}, f_i^{(0)}, s_i\}_{i\in I}$, cf. Sect. 2.5.

For each positive root $\alpha ^{\!\scriptscriptstyle \vee }$ and $r\ge 0$, define the elements $e_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)},f_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)}$ of $Y_\hbar$ via

$$\begin{aligned} \begin{aligned}&e_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)}:=\left[ \left[ \cdots \left[ e_{\alpha ^{\!\scriptscriptstyle \vee }_{i_{\ell }}}^{(r)},e_{\alpha ^{\!\scriptscriptstyle \vee }_{i_{\ell -1}}}^{(0)}\right] ,\cdots , e_{\alpha ^{\!\scriptscriptstyle \vee }_{i_2}}^{(0)}\right] ,e_{\alpha ^{\!\scriptscriptstyle \vee }_{i_1}}^{(0)}\right] ,\\&f_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)}:=\left[ f_{\alpha ^{\!\scriptscriptstyle \vee }_{i_1}}^{(0)},\left[ f_{\alpha ^{\!\scriptscriptstyle \vee }_{i_2}}^{(0)},\cdots ,\left[ f_{\alpha ^{\!\scriptscriptstyle \vee }_{i_{\ell -1}}}^{(0)},f_{\alpha ^{\!\scriptscriptstyle \vee }_{i_{\ell }}}^{(r)}\right] \cdots \right] \right] , \end{aligned} \end{aligned}$$

(A.11)

where $\alpha ^{\!\scriptscriptstyle \vee }=\alpha ^{\!\scriptscriptstyle \vee }_{i_1}+\alpha ^{\!\scriptscriptstyle \vee }_{i_2}+\cdots +\alpha ^{\!\scriptscriptstyle \vee }_{i_{\ell -1}}+\alpha ^{\!\scriptscriptstyle \vee }_{i_\ell }$ is an (ordered) decomposition into simple roots such that the element $[f_{\alpha ^{\!\scriptscriptstyle \vee }_{i_1}},[f_{\alpha ^{\!\scriptscriptstyle \vee }_{i_2}},\cdots ,[f_{\alpha ^{\!\scriptscriptstyle \vee }_{i_{\ell -1}}},f_{\alpha ^{\!\scriptscriptstyle \vee }_{i_{\ell }}}]\cdots ]]$ is a non-zero element of ${\mathfrak {g}}$ (here $\{f_{\alpha ^{\!\scriptscriptstyle \vee }_i}\}_{i\in I}$ denote the standard Chevalley generators of ${\mathfrak {g}}$). We will refer to these elements, together with $\{h_i^{(r)}\}_{i\in I}^{r\in {\mathbb {N}}}$, as the Yangian PBW generators. Throughout this appendix (and the next one), we fix some total ordering on the set of all PBW generators. It is well-known that $Y_\hbar$ is free over ${\mathbb {C}}[\hbar ]$ with a basis given by the PBW monomials, as was proven in Levendorskii (1993). Since the original proof of Levendorskii (1993) contains a significant gap, we give an alternative short proof in Appendix B.

$Y_\hbar$ is a graded Hopf algebra, with $\deg (\hbar )=1$ and $\deg (x^{(r)})=r$ for $x=e_{\alpha ^{\!\scriptscriptstyle \vee }}, h_i, f_{\alpha ^{\!\scriptscriptstyle \vee }}$. Its coproduct is uniquely determined by

$$\begin{aligned} \Delta (x^{(0)})= & {} x^{(0)}\otimes 1+1\otimes x^{(0)}\ \mathrm {for}\ x=e_{\alpha ^{\!\scriptscriptstyle \vee }}, h_i, f_{\alpha ^{\!\scriptscriptstyle \vee }},\nonumber \\ \Delta (s_i)= & {} s_i\otimes 1+1\otimes s_i-\hbar \sum _{\gamma ^{\!\scriptscriptstyle \vee }>0} \langle \alpha _i,\gamma ^{\!\scriptscriptstyle \vee }\rangle f_{\gamma ^{\!\scriptscriptstyle \vee }}^{(0)}\otimes e_{\gamma ^{\!\scriptscriptstyle \vee }}^{(0)}. \end{aligned}$$

(A.12)

A proof of these formulas appears in Guay et al. (2018). Meanwhile, the counit of $Y_\hbar$ is given simply by

$$\begin{aligned} \epsilon \left( e_{\alpha ^{\!\scriptscriptstyle \vee }}^{(0)}\right) =\epsilon \left( f_{\alpha ^{\!\scriptscriptstyle \vee }}^{(0)}\right) =\epsilon \left( h_i^{(0)}\right) =\epsilon (s_i)=0. \end{aligned}$$

(A.13)

Finally, we note that in the classical limit there is an isomorphism of graded Hopf algebras

$$\begin{aligned} Y_\hbar /\hbar Y_\hbar \simeq U({\mathfrak {g}}[t]), \end{aligned}$$

(A.14)

where $U({\mathfrak {g}}[t])$ carries the loop grading.

1.6 The Drinfeld–Gavarini Dual of the Yangian

In this subsection, we describe the Drinfeld–Gavarini dual $Y_\hbar '$, by applying the results of the previous subsections. To this end, we will verify that Assumptions (As1)–(As4) hold for $Y_\hbar$ and its PBW generators. Note that only assumption (As3) remains; the others hold as discussed in Sect. A.4.

Using Eqs. (A.12, A.13), a straightforward calculation shows that:

Lemma A.5

• (1) $\delta _n(x^{(0)})=\left\{ \begin{array}{cl} x, &{} \text {if }n=1 \\ 0, &{} \text {otherwise} \end{array} \right.$,   for any $x = e_{\alpha ^{\!\scriptscriptstyle \vee }}, h_i, f_{\alpha ^{\!\scriptscriptstyle \vee }}$.

• (2) $\delta _n(s_i)=\left\{ \begin{array}{ll} s_i, &{} \text {if } n=1 \\ - \hbar \sum _{\gamma ^{\!\scriptscriptstyle \vee }>0} \langle \alpha _i,\gamma ^{\!\scriptscriptstyle \vee }\rangle f_{\gamma ^{\!\scriptscriptstyle \vee }}^{(0)}\otimes e_{\gamma ^{\!\scriptscriptstyle \vee }}^{(0)}, &{} \text {if } n=2 \\ 0, &{} \text {otherwise} \end{array}. \right.$

Using this lemma, we can now verify assumption (As3):

Lemma A.6

For any PBW generator $x^{(r)}$ of $Y_\hbar$, the element $X^{(r+1)}:=\hbar x^{(r)}$ belongs to the Drinfeld–Gavarini dual $Y_\hbar '$.

Proof

By the previous lemma, we have $\hbar e_{\alpha ^{\!\scriptscriptstyle \vee }}^{(0)}, \hbar h_i^{(0)}, \hbar f_{\alpha ^{\!\scriptscriptstyle \vee }}^{(0)}, \hbar s_i\in Y_\hbar '$. All the PBW generators $x^{(r)}$ can be obtained by taking repeated commutators of these elements, and $X^{(r+1)}$ by repeated application of the operation $a, b\mapsto \tfrac{1}{\hbar } [a,b]$ to the elements $\hbar e_{\alpha ^{\!\scriptscriptstyle \vee }}^{(0)}, \hbar h_i^{(0)}, \hbar f_{\alpha ^{\!\scriptscriptstyle \vee }}^{(0)}$, and $\hbar s_i$. Since $Y_\hbar '$ is closed under this operation, due to Lemma A.1, the claim follows. $\square$

Thus Proposition A.2 applies providing a complete proof of the description of $Y_\hbar '$ given just before Kamnitzer et al. (2014, Theorem 3.5). Note that, as mentioned above, the relations given in Kamnitzer et al. (2014, Theorem 3.5) are incomplete (with the exception of ${\mathfrak {g}}=\mathfrak {sl}_2$). We do not address this issue here, as our methods do not provide a complete set of relations.

Theorem A.7

The Drinfeld–Gavarini dual $Y_\hbar '$ is free over ${\mathbb {C}}[\hbar ]$, with a basis given by the PBW monomials in the elements $X^{(r+1)}$. In particular, $Y_\hbar '\subset Y_\hbar$ is the ${\mathbb {C}}[\hbar ]$-subalgebra generated by the elements $X^{(r+1)}$.

Applying Proposition A.4, we also obtain the Rees algebra description of $Y_\hbar '$ of Finkelberg et al. (2018). In the case of the Yangian, the filtration $F'_\bullet Y_{\hbar =1}$ from (A.10) is known as the Kazhdan filtration.

Corollary A.8

There is a canonical ${\mathbb {C}}[\hbar ]$-algebra isomorphism

$$\begin{aligned} Y_\hbar '\simeq {\text {Rees}}^{F'_\bullet }(Y_{\hbar =1}) \end{aligned}$$

with the Rees algebra of $Y_{\hbar =1}$ with respect to the Kazhdan filtration.

1.7 The RTT Version

Recall the RTT Yangian $Y^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)$ of Sect. 2.1. We refer to the elements $\{t^{(r)}_{ij}\}_{1\le i,j\le n}^{r\ge 1}$ as the PBW generators of $Y^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)$. Fix some total ordering on the set of all PBW generators; this gives rise to the notion of the PBW monomials in $\{t^{(r)}_{ij}\}_{1\le i,j\le n}^{r\ge 1}$.

$Y^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)$ is an ${\mathbb {N}}$-graded Hopf algebra with $\deg (\hbar )=1,\ \deg (t^{(r)}_{ij})=r-1$. Its coproduct $\Delta ^\mathrm {rtt}$ and counit $\epsilon ^\mathrm {rtt}$ are determined explicitly by

$$\begin{aligned} \Delta ^\mathrm {rtt}(T(z))=T(z)\otimes T(z),\ \epsilon ^\mathrm {rtt}(T(z))=I_n. \end{aligned}$$

(A.15)

Moreover, according to Remark 2.2, we have an isomorphism of graded Hopf algebras

$$\begin{aligned} Y^\mathrm {rtt}_{\hbar }(\mathfrak {gl}_n)/\hbar Y^\mathrm {rtt}_{\hbar }(\mathfrak {gl}_n)\simeq U(\mathfrak {gl}_n[t]). \end{aligned}$$

(A.16)

Proposition A.9

The PBW monomials in $\{t^{(r)}_{ij}\}_{1\le i,j\le n}^{r\ge 1}$ form a basis of a free ${\mathbb {C}}[\hbar ]$-module $Y^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)$.

Proof

The proof is similar to that of Theorem B.3 below. First, combining the isomorphism (A.16) with the PBW theorem for $U(\mathfrak {gl}_n[t])$, we immediately see that the PBW monomials span $Y^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)$ over ${\mathbb {C}}[\hbar ]$. To prove the linear independence of the PBW monomials over ${\mathbb {C}}[\hbar ]$, it suffices to verify that their images are linearly independent over ${\mathbb {C}}$ when we specialize $\hbar$ to any nonzero complex number. The latter holds for $\hbar =1$ (and thus for any $\hbar \ne 0$, since all such specializations are isomorphic), due to (Molev 2007, Theorem 1.4.1).

This completes our proof of Proposition A.9. $\square$

The following result provides a new viewpoint towards ${\mathbf {Y}}^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)$ of Definition 2.3:

Theorem A.10

The Drinfeld–Gavarini dual ${Y^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)}'$ is free over ${\mathbb {C}}[\hbar ]$, with a basis given by the PBW monomials in the elements $\hbar t^{(r)}_{ij}$. In particular, ${Y^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)}'={\mathbf {Y}}^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)$.

Proof

This follows from Proposition A.2 once we verify that Assumptions (As1)–(As4) hold for $Y^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)$. Note that only assumption (As3) remains; the others hold as discussed above. The desired inclusion $\hbar t^{(r)}_{ij}\in {Y^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)}'$ follows immediately from (A.15), due to $({\text {id}}-\epsilon ^\mathrm {rtt})(\hbar t^{(r)}_{ij})=\hbar t^{(r)}_{ij}\in \hbar Y^\mathrm {rtt}_\hbar (\mathfrak {gl}_n)$. $\square$

Since the ${\mathbb {C}}[\hbar ]$-algebra isomorphism of Theorem 2.18 is actually an isomorphism of Hopf algebras, we conclude that it gives rise to an isomorphism of the corresponding Drinfeld–Gavarini duals . This provides an alternative computation-free proof of Proposition 2.21.

Remark A.11

Let us compare the above exposition with that of Molev (2007), where an opposite order of reasoning is used. In loc.cit., the author works with the ${\mathbb {C}}$-algebra $Y^\mathrm {rtt}(\mathfrak {gl}_n)$ defined as a common specialization $Y^\mathrm {rtt}(\mathfrak {gl}_n)=Y^\mathrm {rtt}_{\hbar =1}(\mathfrak {gl}_n)={\mathbf {Y}}^\mathrm {rtt}_{\hbar =1}(\mathfrak {gl}_n)$, endowed with two different filtrations $F_\bullet Y^\mathrm {rtt}(\mathfrak {gl}_n), F'_\bullet Y^\mathrm {rtt}(\mathfrak {gl}_n)$ determined by $\deg ^{F_\bullet } (t^{(r)}_{ij})=r, \deg ^{F'_\bullet } (t^{(r)}_{ij})=r-1$ (these notations are opposite to those we used in Sect. A.3). First, in Molev (2007, Corollary 1.4.2), an algebra isomorphism ${\text {gr}}^{F_\bullet } Y^\mathrm {rtt}(\mathfrak {gl}_n)\simeq {\mathbb {C}}[\{t^{(r)}_{ij}\}_{1\le i,j\le n}^{r\ge 1}]$ is proven, and only then an algebra isomorphism ${\text {gr}}^{F'_\bullet } Y^\mathrm {rtt}(\mathfrak {gl}_n)\simeq U(\mathfrak {gl}_n[t])$ is deduced in Molev (2007, Proposition 1.5.2).

1.8 The Shifted Yangian

In Sects. 2.6 and 2.7 , respectively, the algebras ${\mathbf {Y}}_{\mu }$ (for any coweight $\mu$) and ${\mathbf {Y}}'_{\mu }$ (only for a dominant coweight $\mu$) are defined. In this section, we show that these two definitions are equivalent when $\mu$ is dominant. Note that although these definitions were only given in the case of ${\mathfrak {g}}=\mathfrak {sl}_n$, they can be easily extended to any semisimple Lie algebra ${\mathfrak {g}}$ (cf. Kamnitzer et al. 2014; Finkelberg et al. 2018). Till the end of this subsection, we assume that $\mu$ is dominant.

We first recall two auxiliary algebras. The first is the ${\mathbb {C}}$-algebra $Y_\mu$ defined in Sect. 2.6, and the second is the ${\mathbb {C}}[\hbar ]$-algebra $Y_{\mu , \hbar }$ introduced in Sect. 2.7. Both have PBW bases in the corresponding generators over their respective ground rings, by Theorems 2.26 and 2.29 , respectively.

Fixing a splitting $\mu =\mu _1+\mu _2$, recall that $Y_\mu$ has a corresponding filtration $F^\bullet _{\mu _1, \mu _2} Y_\mu$, see (2.23). Similarly $Y_{\mu ,\hbar }$ has a corresponding grading, defined by setting $\deg (\hbar ) =1$ and

$$\begin{aligned} \deg (e_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)})=\alpha ^{\!\scriptscriptstyle \vee }(\mu _1)+r,\quad \deg (f_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)})=\alpha ^{\!\scriptscriptstyle \vee }(\mu _2)+r,\quad \deg (h_i^{(r)})=\alpha ^{\!\scriptscriptstyle \vee }_i(\mu )+r. \end{aligned}$$

Thus for $x=e_{\alpha ^{\!\scriptscriptstyle \vee }}, f_{\alpha ^{\!\scriptscriptstyle \vee }}, h_i$, we have $\deg (x^{(r)})=\deg (x)+r$, where the internal grading $\deg (x)$ is defined via $\deg (e_{\alpha ^{\!\scriptscriptstyle \vee }}) = \alpha ^{\!\scriptscriptstyle \vee }(\mu _1), \deg (f_{\alpha ^{\!\scriptscriptstyle \vee }}) = \alpha ^{\!\scriptscriptstyle \vee }(\mu _2), \deg (h_i) = \alpha ^{\!\scriptscriptstyle \vee }_i(\mu )$.

By comparing their defining relations, it is clear that there is a ${\mathbb {C}}$-algebra isomorphism

(A.17)

On the PBW generators, this isomorphism involves a shift of labels: $x^{(r)} \mapsto X^{(r+1)}$ for $x=e_{\alpha ^{\!\scriptscriptstyle \vee }}, f_{\alpha ^{\!\scriptscriptstyle \vee }}, h_i$. It follows that $Y_\mu$ inherits a second filtration $G_{\mu _1,\mu _2}^\bullet Y_\mu$, coming from the above grading on $Y_{\mu ,\hbar }$. This is analogous to the situation in Sect. A.3: if by abuse of notation we denote the PBW generators of $Y_\mu$ by $x^{(r)}$, then $G^k_{\mu _1, \mu _2} Y_\mu$ is the span of all PBW monomials

$$\begin{aligned} x_1^{(r_1)}\cdots x_\ell ^{(r_\ell )} \end{aligned}$$

(A.18)

with $(\deg (x_1)+r_1)+\cdots +(\deg (x_\ell )+r_\ell )\le k$. Meanwhile, $F^k_{\mu _1,\mu _2} Y_\mu$ is the span of those monomials (A.18) with $(\deg (x_1)+r_1+1)+\cdots +(\deg (x_\ell )+r_\ell +1)\le k$.

In particular, there is an inclusion $F^k_{\mu _1,\mu _2} Y_\mu \subset G^k_{\mu _1,\mu _2} Y_\mu$, hence, an embedding of the Rees algebras

$$\begin{aligned} \mathrm {Rees}^{F^\bullet _{\mu _1,\mu _2}} (Y_\mu ) \ \subset \ \mathrm {Rees}^{G^\bullet _{\mu _1,\mu _2}} (Y_\mu ). \end{aligned}$$

(A.19)

Now on the one hand, ${\mathbf {Y}}_\mu =\mathrm {Rees}^{F^\bullet _{\mu _1,\mu _2}} (Y_\mu )$ by Definition 2.27. On the other hand, since $Y_{\mu , \hbar }$ is free over ${\mathbb {C}}[\hbar ]$, we have $Y_{\mu ,\hbar }\simeq \mathrm {Rees}^{G^\bullet _{\mu _1,\mu _2}} (Y_\mu )$. Explicitly, on PBW generators this isomorphism is defined by

$$\begin{aligned} Y_{\mu , \hbar }\ni x^{(k)} \mapsto \hbar ^{\deg (x)+k} x^{(k)}\in \hbar ^{\deg (x)+k} G^{\deg (x)+k}_{\mu _1,\mu _2} Y_\mu \subset \mathrm {Rees}^{G^\bullet _{\mu _1,\mu _2}} (Y_\mu ). \end{aligned}$$

(A.20)

Altogether, we obtain an injective homomorphism of graded ${\mathbb {C}}[\hbar ]$-algebras ${\mathbf {Y}}_\mu \hookrightarrow Y_{\mu ,\hbar }$.

We can now prove a generalization of Theorem 2.31 for an arbitrary ${\mathfrak {g}}$:

Theorem A.12

For any dominant coweight $\mu$, there is a canonical ${\mathbb {C}}[\hbar ]$-algebra isomorphism ${\mathbf {Y}}_\mu \simeq {\mathbf {Y}}'_\mu$. For any splitting $\mu =\mu _1+\mu _2$, this isomorphism is compatible with the associated gradings on ${\mathbf {Y}}_\mu$ (from the filtration $F^\bullet _{\mu _1, \mu _2} Y_\mu$) and ${\mathbf {Y}}'_\mu$ (as a graded subalgebra of $Y_{\mu ,\hbar }$).

Proof

All that remains is to check that the image of ${\mathbf {Y}}_\mu \hookrightarrow Y_{\mu ,\hbar }$ is precisely ${\mathbf {Y}}'_\mu$. This follows from the “shift” that distinguishes the filtrations $F^k_{\mu _1,\mu _2} Y_\mu$ and $G^k_{\mu _1, \mu _2} Y_\mu$. Indeed, for a monomial $x_1^{(r_1)}\cdots x_\ell ^{(r_\ell )}\in F^k_{\mu _1,\mu _2} Y_\mu$, the corresponding element in the Rees algebra is

$$\begin{aligned} \hbar ^{k} x_1^{(r_1)}\cdots x_\ell ^{(r_\ell )}\in \hbar ^k F^k_{\mu _1,\mu _2} Y_\mu \subset \mathrm {Rees}^{F^\bullet _{\mu _1,\mu _2}} (Y_\mu ). \end{aligned}$$

But in the Rees algebra $\mathrm {Rees}^{G^\bullet _{\mu _1,\mu _2}} (Y_\mu ) \simeq Y_{\mu ,\hbar }$, by inverting (A.20) this element gets sent to

$$\begin{aligned} \hbar ^{k-(\deg (x_1)+r_1)-\cdots -(\deg (x_\ell )+r_\ell )} x_1^{(r_1)}\cdots x_\ell ^{(r_\ell )}\in Y_{\mu ,\hbar }. \end{aligned}$$

Since $(\deg (x_1)+r_1+1)+\cdots +(\deg (x_\ell )+r_\ell +1)\le k$, we can rewrite this as

$$\begin{aligned} \hbar ^{k-(\deg (x_1)+r_1+1)-\cdots -(\deg (x_\ell )+r_\ell +1)} (\hbar x_1^{(r_1)})\cdots (\hbar x_\ell ^{(r_\ell )}), \end{aligned}$$

which lies in ${\mathbf {Y}}'_\mu$. Taking spans of such monomials, we see that ${\mathbf {Y}}_\mu =\mathrm {Rees}^{F^\bullet _{\mu _1,\mu _2}} (Y_\mu )\subset {\mathbf {Y}}'_\mu$. But it is easy to see that the generators of ${\mathbf {Y}}'_\mu$ lie in $\mathrm {Rees}^{F^\bullet _{\mu _1,\mu _2}} (Y_\mu$), so actually ${\mathbf {Y}}_\mu ={\mathbf {Y}}'_\mu$. $\square$

1.9 The Shifted Yangian, Construction III

Motivated by the discussion of the previous subsection, we provide one more alternative definition of the shifted Yangians. Fix a coweight $\mu$ of ${\mathfrak {g}}$ and set $b_i:=\alpha ^{\!\scriptscriptstyle \vee }_i(\mu )$, where $\{\alpha ^{\!\scriptscriptstyle \vee }_i\}_{i\in I}$ are the simple roots of ${\mathfrak {g}}$. Let ${\mathcal {Y}}_{\mu ,\hbar }$ be the associative ${\mathbb {C}}[\hbar ,\hbar ^{-1}]$-algebra generated by $\{e_i^{(r)},f_i^{(r)},h_i^{(s_i)}\}_{i\in I}^{r\ge 0, s_i\ge -b_i}$ with the defining relations similar to those of (2.24) (but generalized to any ${\mathfrak {g}}$) with the only exception:

$$\begin{aligned}{}[e_i^{(r)},f_j^{(r')}]= {\left\{ \begin{array}{ll} h_i^{(r+r')}, &{} \text{ if } i=j\ \mathrm {and}\ r+r'\ge -b_i \\ \hbar ^{-1}, &{} \text{ if } i=j\ \mathrm {and}\ r+r'=-b_i-1 \\ 0, &{} \text{ otherwise } \end{array}\right. }. \end{aligned}$$

(A.21)

Remark A.13

If $\mu$ is dominant, the equality $r+r'=-b_i-1$ never occurs for $r,r'\ge 0$. Thus, ${\mathcal {Y}}_{\mu ,\hbar }$ is the ${\mathbb {C}}[\hbar ,\hbar ^{-1}]$-extension of scalars of $Y_{\mu ,\hbar }$ of Appendix A.7 for dominant $\mu$.

Define the elements $\{e_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)},f_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)}\}_{\alpha ^{\!\scriptscriptstyle \vee }\in \Delta ^+}^{r\ge 0}$ (here $\Delta ^+$ denotes the set of positive roots of ${\mathfrak {g}}$) of ${\mathcal {Y}}_{\mu ,\hbar }$ following (2.25) (but generalized from type A to any ${\mathfrak {g}}$, cf. (Finkelberg et al. 2018, (3.1))). Choose any total ordering on the set of PBW generators as in (2.26) (but generalized from type A to any ${\mathfrak {g}}$, cf. Finkelberg et al. (2018, (3.4))). The following is analogous to Finkelberg et al. (2018, Corollary 3.15) (cf. Theorem 2.26 in type A):

Theorem A.14

For any coweight $\mu$, the PBW monomials form a basis of a free ${\mathbb {C}}[\hbar ,\hbar ^{-1}]$-module ${\mathcal {Y}}_{\mu ,\hbar }$.

This follows immediately from Finkelberg et al. (2018, Corollary 3.15) and the following simple result:

Lemma A.15

Fix a pair of coweights $\mu _1,\mu _2$ such that $\mu _1+\mu _2=\mu$.

(a) There is an isomorphism of ${\mathbb {C}}[\hbar ,\hbar ^{-1}]$-algebras defined by

$$\begin{aligned} h_i^{(r)} \mapsto \hbar ^{\alpha ^{\!\scriptscriptstyle \vee }_i(\mu )+r} H_i^{(r+1)},\ e_i^{(r)} \mapsto \hbar ^{\alpha ^{\!\scriptscriptstyle \vee }_i(\mu _1)+r} E_i^{(r+1)},\ f_i^{(r)} \mapsto \hbar ^{\alpha ^{\!\scriptscriptstyle \vee }_i(\mu _2)+r} F_i^{(r+1)}. \end{aligned}$$

(b) The above isomorphism sends the PBW generators as follows:

$$\begin{aligned} e_{\alpha ^\vee }^{(r)} \mapsto \hbar ^{\alpha ^{\!\scriptscriptstyle \vee }(\mu _1)+r} E_{\alpha ^\vee }^{(r+1)},\ f_{\alpha ^\vee }^{(r)} \mapsto \hbar ^{\alpha ^{\!\scriptscriptstyle \vee }(\mu _2) + r} F_{\alpha ^\vee }^{(r+1)}. \end{aligned}$$

Proof

Part (a) is straightforward. Part (b) follows by comparing (2.21) and (2.25). $\square$

Following Definition 2.30, we introduce:

Definition A.16

Let ${\mathcal {Y}}_{\mu }$ be the ${\mathbb {C}}[\hbar ]$-subalgebra of ${\mathcal {Y}}_{\mu ,\hbar }$ generated by

$$\begin{aligned} \{\hbar e_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)}\}_{\alpha ^{\!\scriptscriptstyle \vee }\in \Delta ^+}^{r\ge 0}\cup \{\hbar f_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)}\}_{\alpha ^{\!\scriptscriptstyle \vee }\in \Delta ^+}^{r\ge 0}\cup \{\hbar h_i^{(s_i)}\}_{i\in I}^{s_i\ge -b_i}. \end{aligned}$$

The following is the main result of this subsection:

Theorem A.17

For any coweight $\mu$, there is a canonical ${\mathbb {C}}[\hbar ]$-algebra isomorphism ${\mathbf {Y}}_\mu \simeq {\mathcal {Y}}_\mu$.

Remark A.18

We note that Theorem A.17 does not imply Theorem A.12, since the algebra ${\mathbf {Y}}'_\mu$ could a priori have an $\hbar$-torsion.

As ${\mathbf {Y}}_\mu =\mathrm {Rees}^{F_{\mu _1,\mu _2}^\bullet } (Y_\mu )$, by the very definition of the Rees algebra we have a natural embedding ${\mathbf {Y}}_\mu \subset Y_\mu [\hbar , \hbar ^{-1}]$. Applying Lemma A.15 with the same decomposition $\mu =\mu _1+\mu _2$, we also obtain an embedding . Therefore, Theorem A.17 follows from:

Lemma A.19

The images of ${\mathbf {Y}}_\mu$ and ${\mathcal {Y}}_\mu$ in $Y_\mu [\hbar , \hbar ^{-1}]$ are equal.

Proof

The filtration $F_{\mu _1,\mu _2}^\bullet Y_\mu$ is defined by the degrees of PBW monomials as in (Finkelberg et al. 2018, (5.1)) (cf. (2.23) in type A). In particular, ${\mathbf {Y}}_\mu$ is the ${\mathbb {C}}[\hbar ]$-subalgebra of $Y_\mu [\hbar , \hbar ^{-1}]$ generated by the elements

$$\begin{aligned} \hbar ^{\alpha ^{\!\scriptscriptstyle \vee }(\mu _1) + r} E_{\alpha ^\vee }^{(r)},\ \hbar ^{\alpha ^{\!\scriptscriptstyle \vee }_i(\mu ) +r} H_i^{(r)},\ \hbar ^{\alpha ^{\!\scriptscriptstyle \vee }(\mu _2) + r} F_{\alpha ^\vee }^{(r)}. \end{aligned}$$

Note that these are precisely the images of the generators $\hbar e_{\alpha ^\vee }^{(r-1)}, \hbar h_i^{(r-1)}, \hbar f_{\alpha ^\vee }^{(r-1)}$ of ${\mathcal {Y}}_\mu$ under the isomorphism of Lemma A.15. The claim follows. $\square$

Remark A.20

We note that Lemma A.19 provides another proof of the fact that the Rees algebras $\mathrm {Rees}^{F^\bullet _{\mu _1,\mu _2}} (Y_\mu )$ are canonically isomorphic for any choice of a splitting $\mu =\mu _1+\mu _2$.

Appendix B: A Short Proof of the PBW Theorem for the Yangians

The PBW theorem for the Yangians is well-known and was first proven by Levendorskii in Levendorskii (1993). However, we feel that the proof of Levendorskii (1993) contains a gap: in Levendorskii (1993, p. 40) it is stated that certain exponents $m^\pm (i,j), m^0(r,j)$ are independent of j without any hint (actually, this seems to be wrong), and this fact plays a crucial role in the proof. For this reason, we present here a short proof of the PBW theorem for the Yangians, which is inspired by Levendorskii’s, but which avoids the aforementioned gap.⁸

1.1 Useful Lemma

Let $A=\bigoplus _{k\in {\mathbb {Z}}} A_k$ be a graded algebra over ${\mathbb {C}}[\hbar ]$, with $1\in A_0$ and $\hbar \in A_1$. Consider its two specializations $A_{\hbar =0}=A/\hbar A$ and $A_{\hbar =1}=A/(\hbar -1) A$. The former is naturally graded via $A_{\hbar =0}=\bigoplus _{k\in {\mathbb {Z}}} A_k/\hbar A_{k-1}$, while the latter inherits a natural filtration $F_{\bullet }A_{\hbar =1}$ with $F_kA_{\hbar =1}$ denoting the image of $\bigoplus _{\ell \le k} A_\ell \subset A$, giving rise to a graded algebra ${\text {gr}} A_{\hbar =1}={\text {gr}}^{F_\bullet } A_{\hbar =1}$.

An explicit relation between the resulting graded ${\mathbb {C}}$-algebras $A_{\hbar =0}$ and ${\text {gr}} A_{\hbar =1}$ is presented in the following result:

Lemma B.1

(a) There is a canonical epimorphism of graded ${\mathbb {C}}$-algebras $\vartheta :A_{\hbar =0}\twoheadrightarrow {\text {gr}} A_{\hbar =1}$.

(b) The kernel of $\vartheta$ is the image of the $\hbar$–torsion⁹ of A in $A_{\hbar =0}$.

Proof

The proof is straightforward. $\square$

1.2 Setup

We follow Sect. A.4 for the conventions regarding the Yangian $Y_\hbar$. However, throughout this section we will work with its specialization $Y_{\hbar =1}$, which we denote simply by Y. Below, we prove the PBW theorem for Y over ${\mathbb {C}}$. We then give a simple argument extending the PBW theorem to the one for $Y_\hbar$ over ${\mathbb {C}}[\hbar ]$. By abuse of notation, we denote the images of the Yangian’s PBW generators in Y by $e_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)}, h_i^{(r)}, f_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)}$.

Let us recall a few basic facts about Y. First of all, there is a natural linear map ${\mathfrak {g}}\rightarrow Y$, defined on the Chevalley generators by $e_i \mapsto e_i^{(0)}, h_i \mapsto h_i^{(0)}, f_i\mapsto f_i^{(0)}$, cf. Lemma 2.12(a). This map is injective. Indeed, according to Drinfeld (1985, Theorem 8),¹⁰ the faithful action of ${\mathfrak {g}}$ on ${\mathfrak {g}}\oplus {\mathbb {C}}$ (the direct sum of the adjoint representation and the trivial one-dimensional) can be extended to an action of Y, hence, any element in the kernel of the above map ${\mathfrak {g}}\rightarrow Y$ is zero.

Second, the grading on $Y_\hbar$ of Sect. A.4 gives rise to a filtration $F_\bullet Y$ as in Sect. B.1. In particular, every PBW generator $x^{(r)}$ belongs to $F_rY$. We note that the coproduct $\Delta :Y \rightarrow Y\otimes Y$ satisfies

$$\begin{aligned} \text {Total filtered degree}\Big (\Delta (x^{(r)})-x^{(r)}\otimes 1-1\otimes x^{(r)}\Big )<r \end{aligned}$$

(B.1)

for any PBW generator $x^{(r)}$, which follows from (A.12). Note that the aforementioned embedding ${\mathfrak {g}}\hookrightarrow Y$ yields a surjection $U({\mathfrak {g}})\twoheadrightarrow F_0 Y$. Moreover, combining the isomorphism (A.14) with Lemma B.1, we obtain a graded algebra epimorphism $\vartheta :U({\mathfrak {g}}[t])\twoheadrightarrow \bigoplus _{k\ge 0}F_k Y / F_{k-1} Y$, in particular, we get a surjective linear map from the degree k part of $U({\mathfrak {g}}[t])$ to $F_k Y / F_{k-1} Y$.

Finally, we recall that there is a translation homomorphism$\tau _a:Y\rightarrow Y[a]$ (here a is a formal parameter) defined on the PBW generators by

$$\begin{aligned} \tau _a(x^{(r)})=\sum _{s=0}^r {r \atopwithdelims ()s}a^{r-s}x^{(s)} \end{aligned}$$

(B.2)

for any PBW generator $x=e_{\alpha ^{\!\scriptscriptstyle \vee }}, h_i, f_{\alpha ^{\!\scriptscriptstyle \vee }}$ (note that this formula is valid for $e_{\alpha ^{\!\scriptscriptstyle \vee }}, f_{\alpha ^{\!\scriptscriptstyle \vee }}$ with $\alpha ^{\!\scriptscriptstyle \vee }$ a non-simple root because of our choices (A.11)). In particular, it follows that the filtered degree of any PBW monomial $y\in Y$ is precisely the degree in a of $\tau _a(y)$.

Define a homomorphism $\Delta _n:Y\rightarrow Y[a_1]\otimes \cdots \otimes Y[a_n]$ as the composition

(B.3)

Here $\Delta ^n$ is the n-th iterated coproduct as in Sect. A.1, and $\tau _{a_i}:Y\rightarrow Y[a_i]$ is the translation homomorphism. In particular, it follows from the above discussion that for any PBW generator $x^{(r)}$, we have

$$\begin{aligned} \Delta _n(x^{(r)})=a_1^r x\otimes 1\otimes \cdots \otimes 1 +1\otimes a_2^r x\otimes 1\otimes \cdots \otimes 1 +\ldots +1\otimes \cdots \otimes 1\otimes a_n^r x \end{aligned}$$

(B.4)

modulo terms of total degree $<r$ in $a_1,\ldots ,a_n$.

1.3 The PBW Theorem for the Yangians

In this subsection, we prove the PBW theorems for Y and $Y_\hbar$.

Theorem B.2

The PBW monomials in the generators $e_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)}, h_i^{(r)}, f_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)}$ form a ${\mathbb {C}}$-basis of Y.

Proof

First, we claim that the PBW monomials span Y. The proof is by induction in the filtered degree. For degree 0, we recall that there is an algebra epimorphism $U({\mathfrak {g}})\twoheadrightarrow F_0 Y$, so the usual PBW theorem for $U({\mathfrak {g}})$ applies; in particular, the PBW monomials in $e_{\alpha ^{\!\scriptscriptstyle \vee }}^{(0)}, h_i^{(0)}, f_{\alpha ^{\!\scriptscriptstyle \vee }}^{(0)}$ span $F_0 Y$. For any $k>0$, recall that the degree k part of $U({\mathfrak {g}}[t])$ surjects onto $F_k Y / F_{k-1} Y$. Combining this with the PBW theorem for $U({\mathfrak {g}}[t])$, we see that $F_k Y$ is spanned by the PBW monomials modulo terms of the lower filtered degree. By induction, the claim follows.

Next, suppose that we could find a relation R between some PBW monomials. Consider the set of the PBW monomials of the maximal filtered degree ${\mathsf {d}}$ that appear non-trivially in this relation. Since this is a finite set, we may find a list of PBW generators $x_1^{({\mathsf {d}}_1)}\le \ldots \le x_n^{({\mathsf {d}}_n)}$ (possibly with multiplicities) such that each of these maximal degree monomials has the form

$$\begin{aligned} (x_1^{({\mathsf {d}}_1)})^{\varepsilon _1}\cdots (x_n^{({\mathsf {d}}_n)})^{\varepsilon _n}, \end{aligned}$$

(B.5)

with all $\varepsilon _i\in \{0,1\}$ and $\sum _i \varepsilon _i {\mathsf {d}}_i={\mathsf {d}}$. When multiplicities do occur, we take the convention that the $\varepsilon _i=1$ appear to the left of $\varepsilon _i=0$. With this convention, each tuple $(\varepsilon _1,\ldots ,\varepsilon _n)$ corresponds uniquely to a PBW monomial.

By (B.4), we find that $\Delta _n(R)$ is a sum of expressions of the form

$$\begin{aligned} \left( \sum _{i=1}^n 1^{\otimes (i-1)}\otimes a_i^{{\mathsf {d}}_1} x^{(0)}_1\otimes 1^{\otimes (n-i)}\right) ^{\varepsilon _1}\cdots \left( \sum _{i=1}^n 1^{\otimes (i-1)}\otimes a_i^{{\mathsf {d}}_n} x^{(0)}_n\otimes 1^{\otimes (n-i)}\right) ^{\varepsilon _n}, \end{aligned}$$

(B.6)

modulo terms of total degree $<{\mathsf {d}}$ in $a_1,\ldots ,a_n$. In particular, in the expression (B.6) there is a summand

$$\begin{aligned} (a_1^{{\mathsf {d}}_1} x^{(0)}_1)^{\varepsilon _1}\otimes (a_2^{{\mathsf {d}}_2} x^{(0)}_2)^{\varepsilon _2}\otimes \cdots \otimes (a_n^{{\mathsf {d}}_n} x^{(0)}_n)^{\varepsilon _n} \end{aligned}$$

(B.7)

which appears with coefficient 1. Moreover, there is a unique PBW monomial for which (B.7) appears as a summand.

Since $x^{(0)}_r$ are in the image of the embedding ${\mathfrak {g}}\hookrightarrow Y$, the elements (B.7) are linearly independent in $Y[a_1]\otimes \cdots \otimes Y[a_n]$. Thus the expressions (B.6) are also linearly independent. This implies that the top total degree term in $\Delta _n(R)$ must be zero, a contradiction.

Hence no linear relations exist, proving the PBW theorem for Y. $\square$

This PBW theorem can be easily generalized to $Y_\hbar$ over ${\mathbb {C}}[\hbar ]$:

Theorem B.3

$Y_\hbar$ is free over ${\mathbb {C}}[\hbar ]$, with a basis of the PBW monomials in the generators $e_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)}, h_i^{(r)}, f_{\alpha ^{\!\scriptscriptstyle \vee }}^{(r)}$.

Proof

Similarly to the proof of Theorem B.2, we see that the PBW monomials span $Y_\hbar$ over ${\mathbb {C}}[\hbar ]$. Moreover, if we specialize $\hbar$ to any complex number, the images of the PBW monomials form a basis. Indeed, the previous theorem proves this for $\hbar =1$ (and thus for any $\hbar \ne 0$, since all such specializations are isomorphic), while the case $\hbar =0$ follows from (A.14) and the PBW theorem for $U({\mathfrak {g}}[t])$.

Suppose that there is some linear relation among the PBW monomials. Its coefficients are elements of ${\mathbb {C}}[\hbar ]$. But they must vanish wherever $\hbar$ is specialized in ${\mathbb {C}}$, since the PBW monomials become a basis. Therefore, all the coefficients are zero. So there are no relations, and the theorem is proved. $\square$