The Huang Algebra Ideal and the Diagonal Shift Property

TL;DR

This paper investigates the algebraic structure of Huang's ideal in a vertex algebra context, proving a key property for Huang's elements and disproving a conjecture through explicit counterexamples.

Contribution

We prove that Huang's elements satisfy the diagonal shift property and disprove Huang's conjecture by constructing elements that do not satisfy this property.

Findings

Huang's elements satisfy the diagonal shift property.

Counterexamples show Huang's conjecture is false in the rank one Heisenberg case.

Abstract

Let $V$ be a grading-restricted vertex algebra and let $A^\infty(V)=U^\infty(V)/Q^\infty(V)$ be the associative algebra constructed by Huang, where $U^\infty(V)$ is the space of column-finite infinite matrices with entries in V and $Q^\infty(V)$ is an ideal of a (nonassociative) algebra structure on $U^\infty(V)$ defined by Huang. Huang introduced families of elements in $Q^\infty(V)$ and conjectured that these elements generate $Q^\infty(V)$. We discover and prove that Huang's elements all satisfy what we call ``the diagonal shift property". On the other hand, in the case that $V$ is the rank one Heisenberg vertex operator algebra, we construct infinitely many linearly independent elements in $Q^\infty(V)$ that do not satisfy the diagonal shift property. As a corollary, we disprove Huang's conjecture.