The Huang Algebra Ideal and the Diagonal Shift Property

Let $V$ be a grading-restricted vertex algebra and let $A^{\mathrm{\infty }}(V)=U^{\mathrm{\infty }}(V)\mathrm{/}{Q}^{\mathrm{\infty }}(V)$ be the associative algebra constructed by Huang, where $U^{\mathrm{\infty }}(V)$ is the space of column-finite infinite matrices with entries in V and $Q^{\mathrm{\infty }}(V)$ is an ideal of a (nonassociative) algebra structure on $U^{\mathrm{\infty }}(V)$ defined by Huang. Huang introduced families of elements in $Q^{\mathrm{\infty }}(V)$ and conjectured that these elements generate $Q^{\mathrm{\infty }}(V)$. 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 $Q^{\mathrm{\infty }}(V)$ that do not satisfy the diagonal shift property. As a corollary, we disprove Huang’s conjecture.