Generalized finite and affine $W$-algebras in type $A$

Choi, Dong Jun; Molev, Alexander; Suh, Uhi Rinn

Generalized finite and affine $W$ -algebras in type $A$

Dec 30, 2024

29 pages

e-Print:

2501.00271 [math-ph]

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Abstract: (arXiv)

We construct a new family of affine

W

-algebras

W^k(\lambda,\mu)

parameterized by partitions

\lambda

and

\mu

associated with the centralizers of nilpotent elements in

\mathfrak{gl}_N

. The new family unifies a few known classes of

W

-algebras. In particular, for the column-partition

\lambda

we recover the affine

W

-algebras

W^k(\mathfrak{gl}_N,f)

of Kac, Roan and Wakimoto, associated with nilpotent elements

f\in\mathfrak{gl}_N

of type

\mu

. Our construction is based on a version of the BRST complex of the quantum Drinfeld-Sokolov reduction. We show that the application of the Zhu functor to the vertex algebras

W^k(\lambda,\mu)

yields a family of generalized finite

W

-algebras

U(\lambda,\mu)

which we also describe independently as associative algebras.

Note:

29 pages

References(23)

Figures(0)

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Generalized finite and affine WWW-algebras in type AAA

Citations per year

Perspectives in Lie theory

W symmetry in conformal field theory

The Models of Two-Dimensional Conformal Quantum Field Theory with Z(n) Symmetry

Vertex algebras and algebraic curves

Center at the critical level for centralizers in type A

Classical W\mathcal {W}W-algebras for Centralizers

Yangian realizations from finite W algebras

Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory

Generalized finite and affine $W$ -algebras in type $A$

Classical $\mathcal {W}$ -algebras for Centralizers