Associative algebras and universal enveloping algebras associated to VOAs (Q2201081)

Let \((V, Y, \mathbf{1}, \omega)\) be a vertex operator algebra. If \(g\) is an isomorphism of \(V\) of finite order \(T\), then an admissible \(g\)-twisted \(V\)-module \(M\) is a \((1/T)\mathbb{Z}_+\)-graded module \(M = \bigoplus_{n \in (1/T)\mathbb{Z}_+} M_i\) equipped with a map \(V \to \text{End}[[z^{1/T}, z^{-1/T}]]\) satisfying some relations of compatibility with \(g\). An associative algebra \(A_{g,n}(V)\), with \(n\) in \((1/T)\mathbb{Z}_+\), can be defined as a quotient of \(V\) in such a way to generalize the Zhu algebra \(A(V)\) [\textit{Y. Zhu}, J. Am. Math. Soc. 9, No. 1, 237--302 (1996; Zbl 0854.17034)] and is proved to be an effective tool in the classification of irreducible admissible \(g\)-twisted \(V\)-modules. For example, if \(M \) is an admissible \(g\)-twisted \(V\)-module then there is a natural representation of \(A_{g,n}(V)\) on \(\bigoplus_{0 \leq i \leq n, i \in (1/T)\mathbb{Z}}M_i\). Therefore, it is not surprising to discover the algebra \(A_{g,n}(V)\) to be isomorphic to some quotient of \(U(V[g])_0\), the subring of \(0\)-homogeneous elements of the universal enveloping algebra \(U(V[g])\) of \(V\) with respect to \(g\). In the twisted case, \(U(V[g])\) is defined as a \((1/T)\mathbb{Z}\)-graded universal algebra of some \((1/T)\mathbb{Z}\)-graded Lie algebra \(\hat{ V }[g]\). The existence of this isomorphism is the main theorem of this work, and the proof relies on a natural explicit construction similar to the one provided in the untwisted case [\textit{X. He}, J. Algebra 491, 265--279 (2017; Zbl 1420.17029)].