On semisimplicity of module categories for finite non-zero index vertex operator subalgebras

Robert McRae Tsinghua University

Category Theory Mathematical Physics Quantum Algebra Representation Theory mathscidoc:2204.04009

Letters in Mathematical Physics, 112, (2), 25, 2022.4
Let V⊆A be a conformal inclusion of vertex operator algebras and let C be a category of grading-restricted generalized V-modules that admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. We give conditions under which C inherits semisimplicity from the category of grading-restricted generalized A-modules in C, and vice versa. The most important condition is that A be a rigid V-module in C with non-zero categorical dimension, that is, we assume the index of V as a subalgebra of A is finite and non-zero. As a consequence, we show that if A is strongly rational, then V is also strongly rational under the following conditions: A contains V as a V-module direct summand, V is C_2-cofinite with a rigid tensor category of modules, and A has non-zero categorical dimension as a V-module. These results are vertex operator algebra interpretations of theorems proved for general commutative algebras in braided tensor categories. We also generalize these results to the case that A is a vertex operator superalgebra.
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@inproceedings{robert2022on,
  title={On semisimplicity of module categories for finite non-zero index vertex operator subalgebras},
  author={Robert McRae},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220415174205871333057},
  booktitle={Letters in Mathematical Physics},
  volume={112},
  number={2},
  pages={25},
  year={2022},
}
Robert McRae. On semisimplicity of module categories for finite non-zero index vertex operator subalgebras. 2022. Vol. 112. In Letters in Mathematical Physics. pp.25. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220415174205871333057.
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