2008/11/01 by Yi-Zhi Huang, YI-ZHI HUANG · 4 citations
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebra representation #Algebraic structures and combinatorial models #Categorical variable #Combinatorics #Conjecture #Graph #Homotopy and Cohomology in Algebraic Topology #Isomorphism (crystallography) #Mathematics #Pure mathematics #Tensor (intrinsic definition) #Tensor contraction #Tensor product #Tensor product of Hilbert spaces #Tensor product of modules #Vertex (graph theory) #Vertex operator algebra
paper · doi:10.1142/s0219199708003083
crossref issued 2008/11/01 · crossref published 2008/11/01 · crossref published-print 2008/11/01 · openalex publication_date 2008/11/01 · crossref created 2008/11/19 · crossref published-online 2011/11/20 · crossref deposited 2020/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/09 · crossref indexed 2026/08/05
Let V be a simple vertex operator algebra satisfying the following conditions: (i) V (n) = 0 for n < 0, V (0) = ℂ1 and V′ is isomorphic to V as a V-module. (ii) Every ℕ-gradable weak V-module is completely reducible. (iii) V is C 2 -cofinite. (In the presence of Condition (i), Conditions (ii) and (iii) are equivalent to a single condition, namely, that every weak V-module is completely reducible.) Using the results obtained by the author in the formulation and proof of the general version of the Verlinde conjecture and in the proof of the Verlinde formula, we prove that the braided tensor category structure on the category of V-modules is rigid, balanced and nondegenerate. In particular, the category of V-modules has a natural structure of modular tensor category. We also prove that the tensor-categorical dimension of an irreducible V-module is the reciprocal of a suitable matrix element of the fusing isomorphism under a suitable basis.