Rigidity of non-negligible objects of moderate growth in braided categories
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914334418927616 |
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| author | Etingof, Pavel Penneys, David |
| author_facet | Etingof, Pavel Penneys, David |
| contents | Let $k$ be a field, and let $\mathcal{C}$ be a Cauchy complete $k$-linear braided category with finite dimensional morphism spaces and ${{\rm End}(\bf 1)}=k$. We call an indecomposable object $X$ of $\mathcal C$ non-negligible if there exists $Y\in \mathcal{C}$ such that $\bf 1$ is a direct summand of $Y\otimes X$. We prove that every non-negligible object $X\in \mathcal{C}$ such that ${\rm dim}{\rm End}(X^{\otimes n})<n!$ for some $n$ is automatically rigid. In particular, if $\mathcal{C}$ is semisimple of moderate growth and weakly rigid, then $\mathcal{C}$ is rigid. As applications, we simplify Huang's proof of rigidity of representation categories of certain vertex operator algebras, and we get that for a finite semisimple monoidal category $\mathcal{C}$, the data of a $\mathcal{C}$-modular functor is equivalent to a modular fusion category structure on $\mathcal{C}$, answering a question of Bakalov and Kirillov. Finally, we show that if $\mathcal{C}$ is rigid and has moderate growth, then the quantum trace of any nilpotent endomorphism in $\mathcal{C}$ is zero. Hence $\mathcal{C}$ admits a semisimplification, which is a semisimple braided tensor category of moderate growth. Finally, we discuss rigidity in braided r-categories which are not semisimple, which arise in logarithmic conformal field theory. These results allow us to simplify a number of arguments of Kazhdan and Lusztig. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17681 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigidity of non-negligible objects of moderate growth in braided categories Etingof, Pavel Penneys, David Quantum Algebra Category Theory Representation Theory Let $k$ be a field, and let $\mathcal{C}$ be a Cauchy complete $k$-linear braided category with finite dimensional morphism spaces and ${{\rm End}(\bf 1)}=k$. We call an indecomposable object $X$ of $\mathcal C$ non-negligible if there exists $Y\in \mathcal{C}$ such that $\bf 1$ is a direct summand of $Y\otimes X$. We prove that every non-negligible object $X\in \mathcal{C}$ such that ${\rm dim}{\rm End}(X^{\otimes n})<n!$ for some $n$ is automatically rigid. In particular, if $\mathcal{C}$ is semisimple of moderate growth and weakly rigid, then $\mathcal{C}$ is rigid. As applications, we simplify Huang's proof of rigidity of representation categories of certain vertex operator algebras, and we get that for a finite semisimple monoidal category $\mathcal{C}$, the data of a $\mathcal{C}$-modular functor is equivalent to a modular fusion category structure on $\mathcal{C}$, answering a question of Bakalov and Kirillov. Finally, we show that if $\mathcal{C}$ is rigid and has moderate growth, then the quantum trace of any nilpotent endomorphism in $\mathcal{C}$ is zero. Hence $\mathcal{C}$ admits a semisimplification, which is a semisimple braided tensor category of moderate growth. Finally, we discuss rigidity in braided r-categories which are not semisimple, which arise in logarithmic conformal field theory. These results allow us to simplify a number of arguments of Kazhdan and Lusztig. |
| title | Rigidity of non-negligible objects of moderate growth in braided categories |
| topic | Quantum Algebra Category Theory Representation Theory |
| url | https://arxiv.org/abs/2412.17681 |