Continuous Tambara-Yamagami tensor categories
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909542236815360 |
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| author | Marín-Salvador, Adrià |
| author_facet | Marín-Salvador, Adrià |
| contents | We present a new model for continuous tensor categories as algebra objects in the Morita bicategory of $\mathrm{C}^*$-algebras. In this setting, we generalize the construction of Tambara-Yamagami tensor categories from finite abelian groups to locally compact abelian groups, and provide a classification of continuous Tambara-Yamagami tensor categories for a locally compact group $G$. A continuous Tambara-Yamagami tensor category associated to a locally compact group $G$ is a continuous tensor category that has a single non-invertible simple object $τ$ such that $τ\otimes τ$ decomposes as a direct integral indexed over $G$, meaning $τ\otimesτ\cong L^2(G)$. We show that continuous Tambara-Yamagami tensor categories for $G$ are classified by a continuous symmetric nondegenerate bicharacter $χ: G\times G\to U(1)$ and a sign $ξ\in\{\pm 1\}$. We also prove that, if a $\mathrm{W}^*$-tensor category $\mathcal{C}$ obeys the Tambara-Yamagami fusion rules, then its associators are automatically continuous in the sense that $\mathcal{C}$ is obtained from a continuous tensor category by forgetting its topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_14596 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continuous Tambara-Yamagami tensor categories Marín-Salvador, Adrià Quantum Algebra High Energy Physics - Theory Category Theory Operator Algebras We present a new model for continuous tensor categories as algebra objects in the Morita bicategory of $\mathrm{C}^*$-algebras. In this setting, we generalize the construction of Tambara-Yamagami tensor categories from finite abelian groups to locally compact abelian groups, and provide a classification of continuous Tambara-Yamagami tensor categories for a locally compact group $G$. A continuous Tambara-Yamagami tensor category associated to a locally compact group $G$ is a continuous tensor category that has a single non-invertible simple object $τ$ such that $τ\otimes τ$ decomposes as a direct integral indexed over $G$, meaning $τ\otimesτ\cong L^2(G)$. We show that continuous Tambara-Yamagami tensor categories for $G$ are classified by a continuous symmetric nondegenerate bicharacter $χ: G\times G\to U(1)$ and a sign $ξ\in\{\pm 1\}$. We also prove that, if a $\mathrm{W}^*$-tensor category $\mathcal{C}$ obeys the Tambara-Yamagami fusion rules, then its associators are automatically continuous in the sense that $\mathcal{C}$ is obtained from a continuous tensor category by forgetting its topology. |
| title | Continuous Tambara-Yamagami tensor categories |
| topic | Quantum Algebra High Energy Physics - Theory Category Theory Operator Algebras |
| url | https://arxiv.org/abs/2503.14596 |