Hecke-Clifford algebras at roots of unity and conformal embeddings
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916682638819328 |
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| author | Edie-Michell, Cain Wenzl, Hans |
| author_facet | Edie-Michell, Cain Wenzl, Hans |
| contents | In this paper we give a combinatorial description of the Cauchy completion of the categories $\mathcal{E}_q$ and $\overline{\mathcal{SE}_N}$ recently introduced by the first author and Snyder. This in turns gives a combinatorial description of the categories $\overline{\operatorname{Rep}(U_q(\mathfrak{sl}_N))}_{A}$ where $A$ is the ètale algebra object corresponding to the conformal embedding $\mathfrak{sl}_N$ level $N$ into $\mathfrak{so}_{N^2-1}$ level 1. In particular we give a classification of the simple objects of these categories, a formula for their quantum dimensions, and fusion rules for tensoring with the defining object. Our method of obtaining these results is the Schur-Weyl approach of studying the representation theory of certain endomorphism algebras in $\mathcal{E}_q$ and $\mathcal{SE}_N$, which are known to be subalgebras of Hecke-Clifford algebras. We build on existing literature to study the representation theory of the Hecke-Clifford algebras at roots of unity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_07201 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hecke-Clifford algebras at roots of unity and conformal embeddings Edie-Michell, Cain Wenzl, Hans Quantum Algebra Rings and Algebras Representation Theory In this paper we give a combinatorial description of the Cauchy completion of the categories $\mathcal{E}_q$ and $\overline{\mathcal{SE}_N}$ recently introduced by the first author and Snyder. This in turns gives a combinatorial description of the categories $\overline{\operatorname{Rep}(U_q(\mathfrak{sl}_N))}_{A}$ where $A$ is the ètale algebra object corresponding to the conformal embedding $\mathfrak{sl}_N$ level $N$ into $\mathfrak{so}_{N^2-1}$ level 1. In particular we give a classification of the simple objects of these categories, a formula for their quantum dimensions, and fusion rules for tensoring with the defining object. Our method of obtaining these results is the Schur-Weyl approach of studying the representation theory of certain endomorphism algebras in $\mathcal{E}_q$ and $\mathcal{SE}_N$, which are known to be subalgebras of Hecke-Clifford algebras. We build on existing literature to study the representation theory of the Hecke-Clifford algebras at roots of unity. |
| title | Hecke-Clifford algebras at roots of unity and conformal embeddings |
| topic | Quantum Algebra Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2504.07201 |