Higher Verlinde Categories: The Mixed Case
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912030023221248 |
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| author | Décoppet, Thibault D. |
| author_facet | Décoppet, Thibault D. |
| contents | Over a field of characteristic $p>0$, the higher Verlinde categories $\mathrm{Ver}_{p^n}$ are obtained by taking the abelian envelope of quotients of the category of tilting modules for the algebraic group $\mathrm{SL}_2$. These symmetric tensor categories have been introduced in arXiv:2003.10499 & arXiv:2003.10105, and their properties have been extensively studied in the former reference. In arXiv:2105.07724, the above construction for $\mathrm{SL}_2$ has been generalized to Lusztig's quantum group for $\mathfrak{sl}_2$ and root of unity $ζ$, which produces the mixed higher Verlinde categories $\mathrm{Ver}^ζ_{p^{(n)}}$. Inspired by the results of arXiv:2003.10499, we study the properties of these braided tensor categories in detail. In particular, we establish a Steinberg tensor product formula for the simple objects of $\mathrm{Ver}^ζ_{p^{(n)}}$, construct a braided embedding $\mathrm{Ver}_{p^n}\hookrightarrow \mathrm{Ver}^ζ_{p^{(n+1)}}$, compute the symmetric center of $\mathrm{Ver}^ζ_{p^{(n)}}$, and identify its Grothendieck ring. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20211 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher Verlinde Categories: The Mixed Case Décoppet, Thibault D. Representation Theory Category Theory Quantum Algebra 18M05, 18M15 (Primary), 20G42 (Secondary) Over a field of characteristic $p>0$, the higher Verlinde categories $\mathrm{Ver}_{p^n}$ are obtained by taking the abelian envelope of quotients of the category of tilting modules for the algebraic group $\mathrm{SL}_2$. These symmetric tensor categories have been introduced in arXiv:2003.10499 & arXiv:2003.10105, and their properties have been extensively studied in the former reference. In arXiv:2105.07724, the above construction for $\mathrm{SL}_2$ has been generalized to Lusztig's quantum group for $\mathfrak{sl}_2$ and root of unity $ζ$, which produces the mixed higher Verlinde categories $\mathrm{Ver}^ζ_{p^{(n)}}$. Inspired by the results of arXiv:2003.10499, we study the properties of these braided tensor categories in detail. In particular, we establish a Steinberg tensor product formula for the simple objects of $\mathrm{Ver}^ζ_{p^{(n)}}$, construct a braided embedding $\mathrm{Ver}_{p^n}\hookrightarrow \mathrm{Ver}^ζ_{p^{(n+1)}}$, compute the symmetric center of $\mathrm{Ver}^ζ_{p^{(n)}}$, and identify its Grothendieck ring. |
| title | Higher Verlinde Categories: The Mixed Case |
| topic | Representation Theory Category Theory Quantum Algebra 18M05, 18M15 (Primary), 20G42 (Secondary) |
| url | https://arxiv.org/abs/2407.20211 |