Asymptotic invariants for fusion algebras associated with compact quantum groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912677288214528 |
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| author | Krajczok, Jacek Skalski, Adam |
| author_facet | Krajczok, Jacek Skalski, Adam |
| contents | We introduce and study certain asymptotic invariants associated with fusion algebras (equipped with a dimension function), which arise naturally in the representation theory of compact quantum groups. Our invariants generalise the analogous concepts studied for classical discrete groups. Specifically we introduce uniform Følner constants and the uniform Kazhdan constant for a regular representation of a fusion algebra, and establish a relationship between these, amenability, and the exponential growth rate considered earlier by Banica and Vergnioux. Further we compute the invariants for fusion algebras associated with % discrete duals of quantum $SU_q(2)$ and $SO_q(3)$ and determine the uniform exponential growth rate for the fusion algebras of all $q$-deformations of semisimple, simply connected, compact Lie groups and for all free unitary quantum groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_14081 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic invariants for fusion algebras associated with compact quantum groups Krajczok, Jacek Skalski, Adam Operator Algebras Quantum Algebra Primary 46L65, Secondary 18D10, 20F69, 20G42, 46L89 We introduce and study certain asymptotic invariants associated with fusion algebras (equipped with a dimension function), which arise naturally in the representation theory of compact quantum groups. Our invariants generalise the analogous concepts studied for classical discrete groups. Specifically we introduce uniform Følner constants and the uniform Kazhdan constant for a regular representation of a fusion algebra, and establish a relationship between these, amenability, and the exponential growth rate considered earlier by Banica and Vergnioux. Further we compute the invariants for fusion algebras associated with % discrete duals of quantum $SU_q(2)$ and $SO_q(3)$ and determine the uniform exponential growth rate for the fusion algebras of all $q$-deformations of semisimple, simply connected, compact Lie groups and for all free unitary quantum groups. |
| title | Asymptotic invariants for fusion algebras associated with compact quantum groups |
| topic | Operator Algebras Quantum Algebra Primary 46L65, Secondary 18D10, 20F69, 20G42, 46L89 |
| url | https://arxiv.org/abs/2502.14081 |