RIGIDITY AND MODULARITY OF VERTEX TENSOR CATEGORIES

By Yi-Zhi Huang. 2005

253 citations recorded by the source corpus.

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 C2-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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9674dcefa7b0d3778d750e9cee1bfeb0e0851db0

How researchers see this paper

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In Algebra and Number Theory

1. T. Creutzig
2. Lin Zhang
3. Yi-Zhi Huang
4. James Lepowsky
5. Jinwei Yang
6. T. Arakawa
7. A. Kirillov
8. Haisheng Li

In Geometry and Topology

1. T. Creutzig
2. Jinwei Yang
3. James Lepowsky
4. Yi-Zhi Huang
5. Lin Zhang
6. T. Arakawa
7. A. Kirillov
8. V. Ostrik

In Mathematical Physics

1. T. Creutzig
2. Yi-Zhi Huang
3. Jinwei Yang
4. James Lepowsky
5. T. Arakawa
6. Lin Zhang
7. A. Kirillov
8. V. Ostrik

Citations in the corpus

• Braided Tensor Categories of Admissible Modules for Affine Lie Algebras (2017, methodology): “akalov and Kirillov [BK] also gave a construction of this braided tensor category structure (but did not give proofs of the rigidity and the nondegeneracy property as is claimed in the book [BK]). In [H6], using a formula proved and used to derive the Verlinde formula in [H5], the second author prov”
• Fusion categories for affine vertex algebras at admissible levels (2018, methodology): “The Hopf links of interest of Wk(g) are given by Verlinde’s formula [1] via character’s Sχ by Sm (0,λ),(0,μ) S (0,0),(0,0) = Sm (0,λ),(0,μ) S (0,0),(0,μ) Sm (0,0),(0,μ) S (0,0),(0,0) = S (0,λ),(0,μ) S (0,0),(0,μ) S (0,0),(0,μ) S (0,0),(0,0) = S (0,λ),(0,μ) S (0,0),(0,0) = e χμ(e(−(u+ v)/u);λ) χ0(e(−”
• A remark on the C2-cofiniteness condition on vertex algebras (2010, background): “This condition has been assumed in many significant theories of vertex operator algebras, such as [6,16,17,25,26,29].”
• Braided Tensor Categories and Extensions of Vertex Operator Algebras (2014, background): “d 4. In particular, for a vertex operator algebra V satisfying Conditions 5, 6 and 7, the category of V-modules has a natural structure of braided tensor category. The following theorem was proved in [Hu5]: 21 Theorem A.2 Let V be a simple vertex operator algebra satisfying Conditions 5, 6 and 7 abo”
• Logarithmic Tensor Category Theory for Generalized Modules for a Conformal Vertex Algebra, I: Introduction and Strongly Graded Algebras and their Generalized Modules (2010, background): “It was in 2005 that this conjecture was finally proved by the first author in [64] (see also the announcement [61] and the exposition [62]).”