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arXiv · 2501.00271

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

Dong Jun Choi·Alexander Molev·Uhi Rinn Suh
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Abstract

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.

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Dong Jun Choi, Alexander Molev, Uhi Rinn Suh. 2024-12-31. Generalized finite and affine $W$-algebras in type $A$. https://doi.org/10.1093/imrn%2Frnag029

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