TITLE: What is the Zhu algebra of a lattice vertex algebra?
QUESTION [8 upvotes]: Associated to a vertex algebra $V$ is an associative algebra $A(V)$, the Zhu algebra. Its defining property is approximately that the representations of $V$ and of $A(V)$ are the same.
In vertex operator algebras associated to affine and Virasoro algebras, Frenkel and Zhu prove for instance that the Zhu algebra of the affine vertex algebra $V_k(\mathfrak{g})$ is $U(\mathfrak{g})$.
Question: Is the Zhu algebra of the vertex algbera $V_k(L)$ associated to lattice $L$ known?
Writing $\mathfrak{h}=L\otimes_{\mathbf{Z}}\mathbf{C}$, there is a map $V_k(\mathfrak{h})\hookrightarrow V_k(L)$, which gives a map $U(\mathfrak{h})\to A(V_k(L))$.

REPLY [4 votes]: I'm expanding Reimundo Heluani's link, which gives the answer when $L$ is an even positive definite lattice. Write $\mathfrak{h}=L\otimes_{\mathbf{Z}}\mathbf{C}$. Every $\alpha\in L$ gives two elements, $E_\alpha\in \mathbf{C}[L]$ and $\alpha\in\mathfrak{h}$.
The Zhu algebra is
$$A(V_L)\ =\ U(\mathfrak{h})\otimes\mathbf{C}[L]/\ (\alpha -(\alpha,\alpha)/2)E_\alpha.$$
The algebra structure on $ U(\mathfrak{h})$ is the usual one, the structure on $\mathbf{C}[L]$ is almost the usual one
$$E_\alpha\cdot E_\beta\ =\ \text{const.}\cdot E_{\alpha+\beta}$$
(for the constants see equations 2.9 and 2.10 of arxiv.org/abs/q-alg/9605032), which together with
$$[\alpha, E_\beta]\ =\ (\alpha,\beta) E_\beta$$
give the algebra structure on $U(\mathfrak{h})\otimes\mathbf{C}[L]$.
It is a finitely generated algebra, and contains a copy of $U(\mathfrak{h})$ within it.