A Vertex Algebra Commutant for the $\beta\gamma$-System and Howe pairs

TL;DR

This paper constructs a vertex algebra commutant for the $eta ext{-}gamma$-system associated with the adjoint representation of $sl(2, ext{C})$, revealing a new Howe pair of vertex algebras.

Contribution

It explicitly describes the generators of the commutant in the $eta ext{-}gamma$-system and introduces a novel Howe pair of vertex algebras.

Findings

Explicit generators for the commutant are provided.

A new Howe pair of vertex algebras is identified.

The work extends the understanding of vertex algebra commutants in specific systems.

Abstract

Analogue to commutants in the theory of associative algebras, one can construct a new subalgebra of vertex algebra known as a vertex algebra commutant. In this paper, for the adjoint representation $V$ of Lie algebra $sl(2,\C)$, we describe a commutant of $\beta\gamma$- System $S(V)$ by giving its generators, moreover, we get a new Howe pair of vertex algebras.