Vertex operator algebras and associative algebras
Let \(V\) be a vertex operator algebra (VOA). Then there is a one-to-one correspondence between the irreducible admissible \(V\)-modules and the irreducible modules for Zhu's algebra \(A(V)\) [cf. \textit{Y. Zhu}, J. Am. Math. Soc. 9, 237--302 (1996; Zbl 0854.17034)]. When \(V\) is a rational VOA, then \(A(V)\) is a semisimple algebra and the category of the admissible \(V\)-modules is equivalent to the category of \(A(V)\)-modules. In the present paper, the authors construct a series of associative algebras \(A_{n}\) for every \(n \in\mathbb Z_+\) such that \(A_{n} (V)\) is a quotient of \(A_{n+1} (V)\), and \(A_0 = A(V)\). They show that there is a one-to-one correspondence between the irreducible \(A_{n}(V)\)-modules which cannot factor through \(A_{n-1} (V)\) and the irreducible admissible \(V\)-modules. They prove that VOA \(V\) is rational if and only if all \(A_{n} (V)\) are finite-dimensional semisimple algebras. It seems that \(A_n(V)\)-theory can be very useful in the study of induced modules for VOAs, and the representation theory of irrational VOAs.
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