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Zhu’s algebra, the $$C_2$$ algebra, and twisted modules. (English) Zbl 1232.17038
Bergvelt, Maarten (ed.) et al., Vertex operator algebras and related areas. An international conference in honor of Geoffrey Mason’s 60th birthday, Normal, IL, USA, July 7–11, 2008. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4840-1/pbk). Contemporary Mathematics 497, 65-78 (2009).
To any vertex operator algebra (VOA) $$V$$, one can associate two associative algebras, called Zhu’s algebra and the $$C_2$$-algebra of $$V$$ [Y. Zhu, J. Am. Math. Soc. 9, No. 1, 237–302 (1996; Zbl 0854.17034)]. Zhu’s algebra plays an important role in studying the representation theory of $$V$$, and finite-dimensionality of the $$C_2$$-algebra is an important technical condition in proving modular properties for a rational VOA $$V$$.
In the paper under review, the authors study the relationship between these two algebras. They explain how one can think of Zhu’s algebra as a certain deformation of the $$C_2$$-algebra. In general, the dimension of Zhu’s algebra is less or equal to the dimension of the $$C_2$$-algebra. The authors explicitly determine these dimensions for some well-known rational $$C_2$$-cofinite VOAs, such as the Virasoro minimal models, affine $$sl (2)$$ at positive integer levels and lattice VOAs. It turns out that in most cases, these dimensions are equal. In the case when the dimensions are different, the authors explain how this anomaly is connected with the existence of twisted representations of $$V$$.
For the entire collection see [Zbl 1175.17001].

##### MSC:
 17B69 Vertex operators; vertex operator algebras and related structures 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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