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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2403.06663 |
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| _version_ | 1866914709422211072 |
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| author | Mang, Alexander |
| author_facet | Mang, Alexander |
| contents | A resolution $P$ of the counit of the Hopf $\ast$-algebra $\mathcal{O}(U_n^+)$ of representative functions on van Daele and Wang's free unitary quantum group $U_n^+$ in terms of free $\mathcal{O}(U_n^+)$-modules is computed for arbitrary $n$. A different such resolution was recently found by Baraquin, Franz, Gerhold, Kula and Tobolski. While theirs has desirable properties which $P$ lacks, $P$ is still good enough to compute the (previously known) quantum group cohomology and comes instead with an important advantage: $P$ can be arrived at without the clever combination of certain results potentially very particular to $U_n^+$ that enabled the aforementioned authors to find their resolution. Especially, $P$ relies neither on the resolution for $O_n^+$ obtained by Collins, Härtel and Thom nor the one for $SL_2(q)$ found by Hadfield and Krähmer. Rather, as shown in the present article, the recursion defining the Anick resolution of the counit of $\mathcal{O}(U_n^+)$ can be solved in closed form. That suggests a potential strategy for determining the cohomologies of arbitrary easy quantum groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06663 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Anick resolution for the free unitary quantum group Mang, Alexander Quantum Algebra 20G42 A resolution $P$ of the counit of the Hopf $\ast$-algebra $\mathcal{O}(U_n^+)$ of representative functions on van Daele and Wang's free unitary quantum group $U_n^+$ in terms of free $\mathcal{O}(U_n^+)$-modules is computed for arbitrary $n$. A different such resolution was recently found by Baraquin, Franz, Gerhold, Kula and Tobolski. While theirs has desirable properties which $P$ lacks, $P$ is still good enough to compute the (previously known) quantum group cohomology and comes instead with an important advantage: $P$ can be arrived at without the clever combination of certain results potentially very particular to $U_n^+$ that enabled the aforementioned authors to find their resolution. Especially, $P$ relies neither on the resolution for $O_n^+$ obtained by Collins, Härtel and Thom nor the one for $SL_2(q)$ found by Hadfield and Krähmer. Rather, as shown in the present article, the recursion defining the Anick resolution of the counit of $\mathcal{O}(U_n^+)$ can be solved in closed form. That suggests a potential strategy for determining the cohomologies of arbitrary easy quantum groups. |
| title | Anick resolution for the free unitary quantum group |
| topic | Quantum Algebra 20G42 |
| url | https://arxiv.org/abs/2403.06663 |