Coproduct for affine Yangians and parabolic induction for rectangular W-algebras

Abstract

We construct algebra homomorphisms from affine Yangians to the current algebras of rectangular W-algebras both in type A. The construction is given via the coproduct and the evaluation map for the affine Yangians. As a consequence, we show that parabolic inductions for representations of the rectangular W-algebras can be regarded as tensor product representations of the affine Yangians under the homomorphisms. The same method is applicable also to the super-setting.

Appendices

Appendix A

Lemma  A.1 and Proposition  A.2 are statements on the rectangular W-algebra ${\mathcal {W}}^{\kappa }({\mathfrak {g}},f)$ for $n \ge 1$ and for $n \ge 2$, respectively.

Lemma A.1

We have

$$\begin{aligned} \begin{aligned} {[} W_{i,j}^{(2)}(m), W_{p,q}^{(1)}(m') ]&= \Big ( \delta _{pj} W_{i,q}^{(2)}(m+m') - \delta _{iq} W_{p,j}^{(2)}(m+m') \Big ) \\&\quad - m'(l-1) \Big ( \delta _{iq} \alpha W_{p,j}^{(1)}(m+m') + \delta _{pq} W_{i,j}^{(1)}(m+m') \Big ) \\&\quad - \dfrac{m'(m'-1)}{2} \delta _{m+m',0} l(l-1)\alpha \Big ( \delta _{iq}\delta _{jp}\alpha + \delta _{ij}\delta _{pq} \Big ). \end{aligned} \end{aligned}$$

In particular, we have

$$\begin{aligned} {[}W_{i,j}^{(2)}(m),W_{p,q}^{(1)}(0)]= & {} \delta _{pj} W_{i,q}^{(2)}(m) - \delta _{iq} W_{p,j}^{(2)}(m),\\ {[}W_{i,j}^{(2)}(m-1),W_{p,q}^{(1)}(1)]= & {} \Big ( \delta _{pj} W_{i,q}^{(2)}(m) - \delta _{iq} W_{p,j}^{(2)}(m) \Big ) \\&- (l-1) \Big ( \delta _{iq} \alpha W_{p,j}^{(1)}(m) + \delta _{pq} W_{i,j}^{(1)}(m) \Big ). \end{aligned}$$

The following assertion slightly refines Proposition 4.5. It can be regarded as an analog of Proposition 6.3.

Proposition A.2

Assume $l \ge 2,$ $n \ge 2$ and $\alpha \ne 0$. Then, ${\mathfrak {U}}({\mathcal {W}}^{\kappa }({\mathfrak {g}},f))$ is topologically generated by

$$\begin{aligned}&\left\{ W_{n,1}^{(1)}(1),\ W_{1,n}^{(1)}(-1),\ W_{i,i+1}^{(1)}(0),\ W_{i+1,i}^{(1)}(0) \mid i=1,\ldots ,n-1 \right\} \\&\quad \cup \left\{ W_{i,i+1}^{(2)}(0) \mid i=1,\ldots ,n-1 \right\} . \end{aligned}$$

Proof

We will gradually construct the elements $W_{i,j}^{(1)}(m)$, $W_{i,j}^{(2)}(m)$ for $i,j=1,\ldots ,n$ and $m \in {\mathbb {Z}}$. They are topological generators of ${\mathfrak {U}}({\mathcal {W}}^{\kappa }({\mathfrak {g}},f))$ when $n \ge 2$ and $\alpha \ne 0$ by Proposition 4.5. The argument below is essentially the same as the one in [37].

• $W_{i,j}^{(1)}(m)$ for $i \ne j$: they are generated from

  $$\begin{aligned} \left\{ W_{n,1}^{(1)}(1),\ W_{1,n}^{(1)}(-1),\ W_{i,i+1}^{(1)}(0),\ W_{i+1,i}^{(1)}(0) \mid i=1,\ldots ,n-1 \right\} \end{aligned}$$

  by the commutation relation (4.4).

• $W_{i,i}^{(2)}(0) - W_{j,j}^{(2)}(0)$ for $i \ne j$: we have

  $$\begin{aligned} \begin{aligned} {[}W_{i,i+1}^{(2)}(0), W_{i+1,i}^{(1)}(0)] = W_{i,i}^{(2)}(0) - W_{i+1,i+1}^{(2)}(0) \end{aligned} \end{aligned}$$

  for $i=1,\ldots ,n-1$.

• $W_{i,j}^{(2)}(m)$ for $i \ne j$: we have

  $$\begin{aligned} \begin{aligned} {[}W_{i,i}^{(2)}(0) - W_{j,j}^{(2)}(0), W_{i,j}^{(1)}(m)] = 2 W_{i,j}^{(2)}(m) + m(l-1)\alpha W_{i,j}^{(1)}(m) \end{aligned} \end{aligned}$$

  for $i \ne j$.

• $W_{i,i}^{(2)}(m)-W_{j,j}^{(2)}(m)$ for $i \ne j$: we have

  $$\begin{aligned} \begin{aligned} {[}W_{i,j}^{(2)}(m), W_{j,i}^{(1)}(0)] = W_{i,i}^{(2)}(m)-W_{j,j}^{(2)}(m) \end{aligned} \end{aligned}$$

  for $i \ne j$.

• $W_{i,i}^{(1)}(m)$: we have

  $$\begin{aligned} \begin{aligned} {[}W_{j,i}^{(2)}(m-1), W_{i,j}^{(1)}(1)] = W_{j,j}^{(2)}(m)-W_{i,i}^{(2)}(m) - (l-1)\alpha W_{i,i}^{(1)}(m) \end{aligned} \end{aligned}$$

  for $i \ne j$.

• $W_{i,i}^{(2)}(m)$: we have

  $$\begin{aligned}&{[}W_{i,i}^{(2)}(m) - W_{j,j}^{(2)}(m), W_{i,i}^{(2)}(m') - W_{j,j}^{(2)}(m')] \\&\quad = (m'-m)\alpha \Big ( W_{i,i}^{(2)}(m+m') + W_{j,j}^{(2)}(m+m') \Big ) + P \end{aligned}$$

  for $i \ne j$, where P is an element of ${\mathfrak {U}}({\mathcal {W}}^{\kappa }({\mathfrak {g}},f))$ which is generated by $W_{a,b}^{(1)}(m'')$ and $W_{c,d}^{(2)}(m''')$ for various $a,b,c,d,m'',m'''$ with $c \ne d$. Hence, under the assumption $\alpha \ne 0$, we see that all the elements of the form $W_{i,i}^{(2)}(m) + W_{j,j}^{(2)}(m)$ for $i \ne j$ and $m \in {\mathbb {Z}}$ belong to the image of $\Phi _{l}$. Thus, $W_{i,i}^{(2)}(m)$ for any i and m belong to the image of $\Phi _{l}$ since so do $W_{i,i}^{(2)}(m) - W_{j,j}^{(2)}(m)$.

$\square$

Assume $\alpha \ne 0$. Let us prove the surjectivity of $\Phi _{l}$, the latter statement of Theorem 9.2. By (9.1) with the commutation relation (4.4), we see that the image of $\Phi _{l}$ contains $W_{i,j}^{(1)}(m)$ and $W_{i,i}^{(1)}(m)-W_{j,j}^{(1)}(m)$ for $i \ne j$ and $m \in {\mathbb {Z}}$. The image of $\Phi _{l}$ contains $W_{i,i}^{(1)}(0)$ for any i since we have

$$\begin{aligned} \Phi _{l} \left( \sum _{i=0}^{n-1} H_{i,1} + \dfrac{\hbar }{2} \sum _{i=1}^{n-1} i H_{i,0} - \dfrac{\hbar }{2} \sum _{i=0}^{n-1} H_{i,0}^{2} \right) = (-\hbar ) \left( -\alpha W_{n,n}^{(1)}(0) - \dfrac{(l\alpha )^2}{2} \right) .\nonumber \\ \end{aligned}$$

(A.1)

We show that the image of $\Phi _{l}$ contains $W_{i,i}^{(1)}(m)$ for any i and $m \ne 0$. The formula (11.1) for $i = 0$ shows that the image of $\Phi _{l}$ contains

$$\begin{aligned} \begin{aligned} H'&= W_{n,n}^{(2)}(0) - W_{1,1}^{(2)}(0) + W_{n,n}^{(1)}(0) \Big ( W_{1,1}^{(1)}(0) - \alpha \Big ) \\&\quad - \sum _{m' \ge 0} \left( W_{n,n}^{(1)}(-m') W_{n,n}^{(1)}(m') - W_{1,1}^{(1)}(-m'-1) W_{1,1}^{(1)}(m'+1) \right) . \end{aligned} \end{aligned}$$

Hence, the assertion follows from

$$\begin{aligned} {[}H', W_{1,1}^{(1)}(m)-W_{2,2}^{(1)}(m)] = -m\alpha W_{1,1}^{(1)}(m). \end{aligned}$$

(A.2)

If $l=1$, this completes the proof. Suppose $l \ge 2$. By Proposition 9.3 together with the fact that the image of $\Phi _{l}$ contains $W_{i,j}^{(1)}(m)$ for any i, j, m, the image of $\Phi _{l}$ contains

$$\begin{aligned} \left\{ W_{n,1}^{(r)}(1),\ W_{1,n}^{(r)}(-1),\ W_{i,i+1}^{(r)}(0),\ W_{i+1,i}^{(r)}(0) \mid r=1,2 \text { and } i=1,\ldots ,n-1 \right\} . \end{aligned}$$

The proof is complete by Proposition  A.2.

Remark A.3

A proof of the surjectivity of $\Phi _{1} = {{\,\mathrm{ev}\,}}$ was initially given by the first named author in [24] by a different method. The above argument, just computing (A.2), supplies a much simpler proof (the computation of (A.1) for $l=1$ has already appeared in [24]).

Appendix B

Lemma B.1

We have

$$\begin{aligned} \begin{aligned} {[}W_{i,i}^{(2)}(0),W_{j,j}^{(2)}(0)]&= - W_{i,i}^{(2)}(0) + W_{j,j}^{(2)}(0)\\&\quad + \sum _{m \ge 0} \Big ( W_{i,j}^{(2)} (-m) W_{j,i}^{(1)} (m) + W_{j,i}^{(1)} (-m-1) W_{i,j}^{(2)} (m+1) \Big ) \\&\quad - \sum _{m \ge 0} \Big ( W_{j,i}^{(2)} (-m) W_{i,j}^{(1)} (m) + W_{i,j}^{(1)} (-m-1) W_{j,i}^{(2)} (m+1) \Big )\\&\quad + (l-1)\alpha \sum _{m \ge 1} m \Big ( W_{j,i}^{(1)} (-m) W_{i,j}^{(1)} (m) - W_{i,j}^{(1)} (-m) W_{j,i}^{(1)} (m) \Big )\\&\quad + (l-1) \sum _{m \ge 1} m \Big ( W_{i,i}^{(1)} (-m) W_{j,j}^{(1)} (m) - W_{j,j}^{(1)} (-m) W_{i,i}^{(1)} (m) \Big ). \end{aligned} \end{aligned}$$

Lemma B.2

For $i < j$, we have

$$\begin{aligned} \begin{aligned} {[}W_{i,i}^{(2)}(0), A_{j}+B_{j}]&= \sum _{m \ge 0} \Big ( -W_{j,i}^{(2)}(-m) W_{i,j}^{(1)}(m) + W_{j,i}^{(1)}(-m) W_{i,j}^{(2)}(m) \Big )\\&\quad + (l-1)\alpha \sum _{m \ge 1} m W_{j,i}^{(1)}(-m) W_{i,j}^{(1)}(m) \\&\quad + (l-1) \sum _{m \ge 1} m \Big ( W_{i,i}^{(1)}(-m) W_{j,j}^{(1)}(m) - W_{j,j}^{(1)}(-m) W_{i,i}^{(1)}(m) \Big ) \end{aligned} \end{aligned}$$

and

$$\begin{aligned} {[}W_{j,j}^{(2)}(0), A_{i}+B_{i}]= & {} \sum _{m \ge 0} \Big ( -W_{i,j}^{(2)}(-m-1) W_{j,i}^{(1)}(m+1) + W_{i,j}^{(1)}(-m-1) W_{j,i}^{(2)}(m+1) \Big )\\&+ (l-1)\alpha \sum _{m \ge 1} m W_{i,j}^{(1)}(-m) W_{j,i}^{(1)}(m) \\&+ (l-1) \sum _{m \ge 1} m \Big ( W_{j,j}^{(1)}(-m) W_{i,i}^{(1)}(m) - W_{i,i}^{(1)}(-m) W_{j,j}^{(1)}(m) \Big ). \end{aligned}$$

Proof of Proposition11.1

We may assume $i < j$. Then, by Lemma B.1, B.2, and (11.2), we have

$$\begin{aligned} \begin{aligned} {[}D_{i}, D_{j}]&= [W_{i,i}^{(2)}(0), W_{j,j}^{(2)}(0)] - [W_{i,i}^{(2)}(0), A_{j}+B_{j}] + [W_{j,j}^{(2)}(0), A_{i}+B_{i}] \\&\quad + [A_{i}+B_{i},A_{j}+B_{j}] \\&= - W_{i,i}^{(2)}(0) + W_{j,j}^{(2)}(0) + W_{i,j}^{(2)}(0) W_{j,i}^{(1)}(0) - W_{j,i}^{(1)}(0) W_{i,j}^{(2)}(0) \\&= - W_{i,i}^{(2)}(0) + W_{j,j}^{(2)}(0) + [W_{i,j}^{(2)}(0), W_{j,i}^{(1)}(0)]. \end{aligned} \end{aligned}$$

This is equal to 0 by Lemma  A.1. $\square$

Appendix C

The equality (11.2) is deduced from the following. We omit its proof.

Lemma C.1

For $i < j$, we have

$$\begin{aligned} {[}A_{i}, A_{j}]= & {} \sum _{\begin{array}{c} m,m' \ge 0\\ m-m'>0 \end{array}} \sum _{a=1}^{i-1} \Big ( W_{j,a}^{(1)}(-m') W_{i,j}^{(1)}(-m+m') W_{a,i}^{(1)}(m) \\&- W_{i,a}^{(1)}(-m) W_{j,i}^{(1)}(m-m') W_{a,j}^{(1)}(m') \Big ),\\ {[}A_{i}, B_{j}]= & {} 0,\\ {[}B_{i}, A_{j}]= & {} \sum _{m,m' \ge 0} \Bigg ( \sum _{a=1}^{i-1} \Big ( -W_{j,a}^{(1)}(-m') W_{i,j }^{(1)}(-m-1) W_{a,i}^{(1)}(m+m'+1) \\&+ W_{i,a}^{(1)}(-m-m'-1) W_{j,i}^{(1)}(m+1) W_{a,j}^{(1)}(m') \Big )\\&+ \sum _{a=j}^{n} \Big ( -W_{j,a}^{(1)}(-m-m'-1) W_{i,j}^{(1)}(m') W_{a,i}^{(1)}(m+1) \\&+ W_{i,a}^{(1)}(-m-1) W_{j,i}^{(1)}(-m') W_{a,j}^{(1)}(m+m'+1) \Big ) \Bigg )\\&+ \sum _{m \ge 1} m \Big ( -W_{i,i}^{(1)}(-m) W_{j,j}^{(1)}(m) + W_{j,j}^{(1)}(-m) W_{i,i}^{(1)}(-m) \Big ),\\ {[}B_{i}, B_{j}]= & {} \sum _{\begin{array}{c} m,m' \ge 0\\ -m+m' \ge 0 \end{array}} \sum _{a=j}^{n} \Big ( W_{j,a}^{(1)}(-m'-1) W_{i,j}^{(1)}(-m+m') W_{a,i}^{(1)}(m+1) \\&- W_{i,a}^{(1)}(-m-1) W_{j,i}^{(1)}(m-m') W_{a,j}^{(1)}(m'+1) \Big )\\&+ l \sum _{m \ge 1} m \Big ( W_{i,i}^{(1)}(-m) W_{j,j}^{(1)}(m) - W_{j,j}^{(1)}(-m) W_{i,i}^{(1)}(-m) \Big ). \end{aligned}$$