2001/12/06 by Masahiko Miyamoto, Miyamoto, Masahiko, Kenichiro Tanabe +2 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #math.QA
paper · pdf · doi:10.48550/arxiv.math/0112054
16 pages, Latex We added a duality theorem
openalex publication_date 2001/12/06 · arxiv created 2002/02/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let V be a vertex operator algebra and G a finite automorphism group of V. For each g∈ G and nonnegative rational number n∈ \mathbb Z/|g|, a g-twisted Zhu algebra Ag,n(V) plays an important role in the theory of vertex operator algebras, but the given product in Ag,n(V) depends on the eigenspaces of g. We show that there is a uniform definition of products on V and we introduce a G-twisted Zhu algebra AG,n(V) which covers all g-twisted Zhu algebras. Assume that V is simple and let \cal S be a finite set of inequivalent irreducible twisted V-modules which is closed under the action of G. There is a finite dimensional semisimple associative algebra \cal Aα(G,\cal S) for a suitable 2-cocycle naturally determined by the G-action on \cal S. We show that a duality theorem of Schur-Weyl type holds for the actions of \cal Aα(G,\cal S) and VG on the direct sum of twisted V-modules in \cal S as an application of the theory of AG,n(V). It follows as a natural consequence of the result that for any g∈ G every irreducible g-twisted V-module is a completely reducible VG-module.