Affine Yangians and rectangular W -algebras
Ryosuke Kodera
Abstract
We construct algebra homomorphisms from affine Yangians to rectangular W -algebras both in type A. The construction is given via thecoproduct and the evaluation map for the affine Yangians. This is ajoint work with Mamoru Ueda [KU].

1Introduction

Ragoucy-Sorba [RS] and Brundan-Kleshchev [BK] have constructed surjective algebra homomorphisms from the Yangian of gln to the finite W algebras associated with nilpotent elements of rectangular type in glN . Ourmain result (Theorem 3.1) is an affine analog of their construction.
Our work is motivated by the Alday-Gaiotto-Tachikawa (AGT) correspondence for parabolic sheaves on P1 × P1 . Using Brundan-Kleshchev’s
result, Braverman-Feigin-Finkelberg-Rybnikov [BFFR] have constructed anaction of the finite W -algebra on the localized equivariant cohomology ofthe moduli space of flags of sheaves on P1 (They call this result “a finiteanalog of the AGT relation”). We expect that our result can be used to construct an action of the W -algebra on the cohomology of the moduli space ofparabolic sheaves.
Please also see expository papers [Kod1], [Kod2] written in Japanese.

2Yangians and rectangular finite W -algebras

We introduce finite W -algebras in the case g = glN with a nilpotent element
whose Jordan form is of rectangular type.
Let g = glN be the complex general linear Lie algebra consisting of
N × N matrices. We denote by eij the matrix unit whose (i, j)-entry is 1.
We assume N = ln with nonzero positive integers l and n. As vector spaces,
we have an isomorphism
gll ⊗ gln ∼= g = glN ,est ⊗ eij 7→ e(s−1)n+i,(t−1)n+j .

(2.1)

Department of Mathematics and Informatics, Graduate School of Science, Chiba University
kodera@math.s.chiba-u.ac.jp

We define a Z-grading of gll by deg est = t − s and denote by (gll )j its degree
j component. It induces a Z-grading of g by
gj = (gll )j ⊗ gln
using the identification (2.1). In particular, we have g0 ∼= gl⊕ln . We set[s]eij = e(s−1)n+i,(s−1)n+j[s]for s = 1, . . . , l and i, j = 1, . . . , n. Then {eij | i, j = 1, . . . , n} forms aC-basis of the s-th component of g0 ∼= gl⊕ln .
We take a nilpotent element f in g as
OO In O
l−1 XnX.
In . . .
f=
esn+i,(s−1)n+i = 
s=1 i=1OOIn O
This is called rectangular nilpotent since its Jordan type is (ln ) and thecorresponding Young diagram has an n × l rectangular shape. Set
p=Mm=Mgj =j≤0j>0gj =MnM

1≤t≤s≤l i,j=1n
M M

Cesn+i,tn+j ,Cesn+i,tn+j

1≤s<t≤l i,j=1and let χ = (f, −) : m → C be the one-dimensional representation of mwhich corresponds to f via the trace form (, ). Let Iχ be the left ideal ofU (g) generated by {x − χ(x) | x ∈ m}. We have a decomposition
U (g) = U (p) ⊕ Iχ

(2.2)

of a C-vector space.
The finite W -algebra U (g, f ) is defined byU (g, f ) = Endg (U (g)/Iχ )op .
By identifying U (g)/Iχ with U (p) via (2.2), it is regarded as a subalgebraof U (p):U (g, f ) = {x ∈ U (p) | [y, x] ∈ Iχ for all y ∈ m}.
Example 2.1 (n = 1 and l = 2). We consider the following case:
0 0g = gl2 ,f=.
1 0We havep=∗ 0∗ ∗,

m=0 ∗0 0.

Then U (g, f ) is generated by
e11 + e22 ,e11 e22 − e21 + e22 .

These elements are the images of elements
e11 + e22 ,e11 e22 − e21 e12 + e22of the center of U (gl2 ) by the projection to U (p) via (2.2). This correspondence induces an algebra isomorphism between the center of U (gl2 ) andU (g, f ) in this example. More generally, Kostant [Kos] proved that U (g, f )is isomorphic to the center of U (g) in the case n = 1 and an arbitrary l(principal nilpotent case).
Ragoucy-Sorba [RS] and Brundan-Kleshchev [BK] related the finite W algebra U (g, f ) to the Yangian.
Theorem 2.2 (Ragoucy-Sorba [RS], Brundan-Kleshchev [BK]). There exists a surjective algebra homomorphism from the Yangian of gln to U (g, f ).
Let ξ : p → g0 be the natural projection and denote by the same letter theinduced algebra homomorphism U (p) → U (g0 ) ∼= U (gln )⊗l . The restrictionof ξ to U (g, f ) is known to be injective. It is called the Miura map. InExample 2.1, ξ(U (g, f )) is generated by
e11 + e22 ,e11 e22 + e22 .

We define an algebra automorphism η on U (g0 ) by[s][s]η(eij ) = eij + δij n(l − s)for s = 1, . . . , l and i, j = 1, . . . , n. We set
Wlfin = η −1 ξ(U (g, f )).
In Example 2.1, since we have η(e11 e22 ) = (e11 + 1)e22 = e11 e22 + e22 , Wlfin
is generated by
e11 + e22 ,e11 e22 .
Brundan-Kleshchev gave an observation that the homomorphism in Theorem 2.2 is related to the coproduct and the evaluation map for the YangianY (gln ). Here
• the coproduct is an algebra homomorphism ∆ : Y (gln ) → Y (gln )⊗2 ;• the evaluation map is an algebra homomorphism ev : Y (gln ) → U (gln ).
Hence we have the compositeev⊗l ◦∆l−1 : Y (gln ) → U (gln )⊗lof the iterated coproduct ∆l−1 = (∆ ⊗ id⊗(l−2) ) ◦ · · · ◦ (∆ ⊗ id) ◦ ∆ and ev⊗l .
Theorem 2.2 can be restated as the following.
Theorem 2.3 (Brundan-Kleshchev [BK] Section 12). The image of ev⊗l ◦∆l−1
coincides with Wlfin .

3Affine Yangians and rectangular W -algebras

Given a Lie algebra a, let â denote its affine Lie algebra, that is, the onedimensional central extension of a[t, t−1 ]. We write the element xtm (x ∈ aand m ∈ Z) by x(m).
The W -algebra W k (g, f ) is defined from the data g, f as Section 2 andk ∈ C by the BRST cohomology (see [AM] for the definition and constructionof generators). It is a subalgebra of a certain completion U (p̂)comp of theuniversal enveloping algebra of p̂. It also admits the Miura map
ξ : W k (g, f ) → U (ĝ0 )comp ∼= U (ĝln )⊗comp l ,which is injective. We define an algebra automorphism η on U (ĝ0 )comp by[s][s]η(eij (m)) = eij (m) + δm,0 δij (k + N − n)(l − s)for s = 1, . . . , l, i, j = 1, . . . , n, and m ∈ Z. We set
Wl = η −1 ξ(W k (g, f )).
ˆ n ) is defined by generators and relations (seeThe affine Yangian Y (sl[Kod2]). Let us summarize its properties:• it is a two-parameter deformation of the universal enveloping algebraˆ n [s];of the universal central extension of the Lie algebra sl
ˆ n ) → Y (slˆ n )⊗comp 2 ;• it admits the coproduct ∆ : Y (slˆ n ) → U (ĝln )comp .
• it admits the evaluation map ev : Y (slThe compositeˆ n ) → U (ĝln )⊗comp lev⊗l ◦∆l−1 : Y (slis defined in a way similar to Section 2. The following is our main result. Itgives an affine analog of Theorem 2.3.
Theorem 3.1 (Kodera-Ueda [KU]). Let n ≥ 3. The image of ev⊗l ◦∆l−1
is contained in Wl . It coincides with Wl if k + N − n 6= 0.

Acknowledgments
The author was supported by JSPS KAKENHI Grant Number 21K03155.

References[AM]

Tomoyuki Arakawa and Alexander Molev, Explicit generators inrectangular affine W-algebras of type A, Lett. Math. Phys. 107(2017), no. 1, 47–59.

[BFFR] Alexander Braverman, Boris Feigin, Michael Finkelberg, andLeonid Rybnikov, A finite analog of the AGT relation I: Finite W algebras and quasimaps’ spaces, Comm. Math. Phys. 308 (2011),
no. 2, 457–478.
[BK]

Jonathan Brundan and Alexander Kleshchev, Shifted Yangians andfinite W -algebras, Adv. Math. 200 (2006), no. 1, 136–195.

[Kod1] Ryosuke Kodera, Affine Yangians and rectangular W -algebras, Proceedings of Algebra Symposium 2021 (in Japanese).
[Kod2], Coproduct for affine Yangians and parabolic induction forrectangular W -algebras, Proceedings of RIMS Workshop “Recentdevelopments in Combinatorial Representation Theory 2022” (inJapanese).

[Kos]

Bertram Kostant, On Whittaker vectors and representation theory,
Invent. Math. 48 (1978), no. 2, 101–184.

[KU]

Ryosuke Kodera and Mamoru Ueda, Coproduct for affine Yangians
and parabolic induction for rectangular W -algebras, Lett. Math.
Phys. 112 (2022), no. 1, Paper No. 3, 37.

[RS]

Eric Ragoucy and Paul Sorba, Yangian realisations from finite Walgebras, Comm. Math. Phys. 203 (1999), no. 3, 551–572.