On the commutant of the principal subalgebra in the $A_1$ lattice vertex algebra

TL;DR

This paper investigates the structure of the commutant of the principal subalgebra in the $A_1$ lattice vertex algebra, revealing it is infinitely generated with a specific Poisson algebra structure, and establishes a dual pair relationship.

Contribution

It provides an explicit description of the commutant of the principal subalgebra in the $A_1$ lattice vertex algebra, including generators and algebraic properties, outside the usual vertex algebra framework.

Findings

The commutant $C$ is not finitely generated.

Zhu's Poisson algebra of $C$ is isomorphic to an infinite-dimensional algebra.

$W$ and $C$ form a dual pair in $V_{A_1}$.

Abstract

The coset (commutant) construction is a fundamental tool to construct vertex operator algebras from known vertex operator algebras. The aim of this paper is to provide a fundamental example of the commutants of vertex algebras ouside vertex operator algebras. Namely, the commutant $C$ of the principal subalgebra $W$ of the $A_1$ lattice vertex operator algebra $V_{A_1}$ is investigated. An explicit minimal set of generators of $C$, which consists of infinitely many elements and strongly generates $C$, is introduced. It implies that the algebra $C$ is not finitely generated. Furthermore, Zhu's Poisson algebra of $C$ is shown to be isomorphic to an infinite-dimensional algebra $\mathbb{C}[x_1,x_2,\ldots]/(x_ix_j\,|\,i,j=1,2,\ldots)$. In particular, the associated variety of $C$ consists of a point. Moreover, $W$ and $C$ are verified to form a dual pair in $V_{A_1}$.