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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 $U(\mathfrak{h})\to A(V_k(L))$.

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  • $\begingroup$ Shouldn't it be the group algebra of $L^*/L$? $\endgroup$
    – Theo Johnson-Freyd
    Commented Jul 12, 2020 at 1:36
  • $\begingroup$ @TheoJohnson-Freyd I have no idea. If you find a reference, I'd be happy to accept that as an answer. $\endgroup$
    – Pulcinella
    Commented Jul 12, 2020 at 10:37
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    $\begingroup$ arxiv.org/abs/q-alg/9605032 $\endgroup$
    – Reimundo Heluani
    Commented Jul 12, 2020 at 13:17
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    $\begingroup$ sorry @TheoJohnson-Freyd for not being verbose, still at the clinic for the birth of my second baby. That paper computes the Zhu algebra in the even lattice case, it starts with an example in rank one to show it's not something trivial. There are newer results in the odd case that escape my memory, but a quick search through the papers referencing this one should find them. $\endgroup$
    – Reimundo Heluani
    Commented Jul 12, 2020 at 17:14
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    $\begingroup$ In RCFT, there is a 1-1 correspondence between Irr(V) <---> Irr(A(V)), W<---> lowest weight subspace of W (let us call it W_0). Then A(V) is equivalent to \bigoplus End(W_0), where the direct sum is over all (equivalence classes of) irreducibles of V. $\endgroup$
    – Bin Gui
    Commented Jul 12, 2020 at 23:37

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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.

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