Rad HAZU, Matematičke znanosti, Vol. 21 (2017), 75-87.

A NOTE ON THE AFFINE VERTEX ALGEBRA ASSOCIATED TO gl(1|1) AT THE CRITICAL LEVEL AND ITS GENERALIZATIONS

Dražen Adamović

Abstract.   In this note we present an explicit realization of the affine vertex algebra V^cri(gl(1|1)) inside of the tensor product F ⊗ M where F is a fermionic verex algebra and M is a commutative vertex algebra. This immediately gives an alternative description of the center of V^cri(gl(1|1)) as a subalgebra M₀ of M. We reconstruct the Molev-Mukhin formula for the Hilbert-Poincare series of the center of V^cri(gl(1|1)). Moreover, we construct a family of irreducible V^cri(gl(1|1))-modules realized on F and parameterized by χ⁺, χ^- ∈ C((z)). We propose a generalization of V^cri(gl(1|1)) as a critical level version of the super W_1+∞ vertex algebra.

2010 Mathematics Subject Classification.   17B69, 17B67.

Key words and phrases.   Vertex algebras, affine Lie superalgebras, critical level, W-algebras.

DOI: https://doi.org/10.21857/yrvgqtpk89

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