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

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.

15 pages, No figures. This is our first paper of series on this topic