Frobenius subalgebra lattices in tensor categories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909866579197952 |
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| author | Ghosh, Mainak Palcoux, Sebastien |
| author_facet | Ghosh, Mainak Palcoux, Sebastien |
| contents | This paper studies Frobenius subalgebra posets in abelian monoidal categories and shows that, under general conditions--satisfied in all semisimple tensor categories over the complex field--they collapse to lattices through a rigidity invariance perspective. Based on this, we extend Watatani's finiteness theorem for intermediate subfactors by proving that, under a weak positivity assumption--met by all semisimple tensor categories over the complex field--and a compatibility condition--fulfilled by all pivotal ones--the lattices arising from connected Frobenius algebras are finite. We also derive a non-semisimple version via semisimplification. Our approach relies on the concept of a formal angle, and the extension of key results--such as the planar algebraic exchange relation and Landau's theorems--to linear monoidal categories.
Major applications of our findings include a stronger version of the Ino-Watatani result: we show that the finiteness of intermediate C*-algebras holds in a finite-index unital irreducible inclusion of C*-algebras without requiring the simple assumption. Moreover, for a finite-dimensional semisimple Hopf algebra H, we prove that H* is a Frobenius algebra object in Rep(H) and has a finite number of rigid invariant Frobenius subalgebras. Finally, we explore a range of applications, including abstract spin chains, vertex operator algebras and speculations on quantum arithmetic involving the generalization of Ore's theorem, Euler's totient and sigma functions, and RH. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_19876 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Frobenius subalgebra lattices in tensor categories Ghosh, Mainak Palcoux, Sebastien Quantum Algebra Combinatorics Category Theory Operator Algebras Rings and Algebras 18M20, 46L37 (Primary) 05E16, 16T05, 17B69, 18M05, 46L05 (Secondary) This paper studies Frobenius subalgebra posets in abelian monoidal categories and shows that, under general conditions--satisfied in all semisimple tensor categories over the complex field--they collapse to lattices through a rigidity invariance perspective. Based on this, we extend Watatani's finiteness theorem for intermediate subfactors by proving that, under a weak positivity assumption--met by all semisimple tensor categories over the complex field--and a compatibility condition--fulfilled by all pivotal ones--the lattices arising from connected Frobenius algebras are finite. We also derive a non-semisimple version via semisimplification. Our approach relies on the concept of a formal angle, and the extension of key results--such as the planar algebraic exchange relation and Landau's theorems--to linear monoidal categories. Major applications of our findings include a stronger version of the Ino-Watatani result: we show that the finiteness of intermediate C*-algebras holds in a finite-index unital irreducible inclusion of C*-algebras without requiring the simple assumption. Moreover, for a finite-dimensional semisimple Hopf algebra H, we prove that H* is a Frobenius algebra object in Rep(H) and has a finite number of rigid invariant Frobenius subalgebras. Finally, we explore a range of applications, including abstract spin chains, vertex operator algebras and speculations on quantum arithmetic involving the generalization of Ore's theorem, Euler's totient and sigma functions, and RH. |
| title | Frobenius subalgebra lattices in tensor categories |
| topic | Quantum Algebra Combinatorics Category Theory Operator Algebras Rings and Algebras 18M20, 46L37 (Primary) 05E16, 16T05, 17B69, 18M05, 46L05 (Secondary) |
| url | https://arxiv.org/abs/2502.19876 |