*-Hopf algebroids
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| author | Beggs, Edwin Han, Xiao Majid, Shahn |
| author_facet | Beggs, Edwin Han, Xiao Majid, Shahn |
| contents | We introduce a theory of $*$-structures for bialgebroids and Hopf algebroids over a $*$-algebra, defined in such a way that the relevant category of (co)modules is a bar category. We show that if $H$ is a Hopf $*$-algebra then the action Hopf algebroid $A\# H$ associated to a braided-commutative algebra in the category of $H$-crossed modules is a full $*$-Hopf algebroid and the Ehresmann-Schauenburg Hopf algebroid $\mathcal{L}(P,H)$ associated to a Hopf-Galois extension or quantum group principal bundle $P$ with fibre $H$ forms a $*$-Hopf algebroid pair, when the relevant (co)action respects $*$. We also show that Ghobadi's bialgebroid associated to a $*$-differential structure $(Ω^{1},\rm d)$ on $A$ forms a $*$-bialgebroid pair and its quotient in the pivotal case a $*$-Hopf algebroid pair when the pivotal structure is compatible with $*$. We show that when $Ω^1$ is simultaneously free on both sides, Ghobadi's Hopf algebroid is isomorphic to $\mathcal{L}(A\#H,H)$ for a smash product by a certain Hopf algebra $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_21089 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | *-Hopf algebroids Beggs, Edwin Han, Xiao Majid, Shahn Quantum Algebra 16T05, 20G42, 46L87, 58B32 We introduce a theory of $*$-structures for bialgebroids and Hopf algebroids over a $*$-algebra, defined in such a way that the relevant category of (co)modules is a bar category. We show that if $H$ is a Hopf $*$-algebra then the action Hopf algebroid $A\# H$ associated to a braided-commutative algebra in the category of $H$-crossed modules is a full $*$-Hopf algebroid and the Ehresmann-Schauenburg Hopf algebroid $\mathcal{L}(P,H)$ associated to a Hopf-Galois extension or quantum group principal bundle $P$ with fibre $H$ forms a $*$-Hopf algebroid pair, when the relevant (co)action respects $*$. We also show that Ghobadi's bialgebroid associated to a $*$-differential structure $(Ω^{1},\rm d)$ on $A$ forms a $*$-bialgebroid pair and its quotient in the pivotal case a $*$-Hopf algebroid pair when the pivotal structure is compatible with $*$. We show that when $Ω^1$ is simultaneously free on both sides, Ghobadi's Hopf algebroid is isomorphic to $\mathcal{L}(A\#H,H)$ for a smash product by a certain Hopf algebra $H$. |
| title | *-Hopf algebroids |
| topic | Quantum Algebra 16T05, 20G42, 46L87, 58B32 |
| url | https://arxiv.org/abs/2412.21089 |