Quantum jet Hopf algebroids by cotwist
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912784125526016 |
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| author | Han, Xiao Majid, Shahn |
| author_facet | Han, Xiao Majid, Shahn |
| contents | We introduce a cotwist construction for Hopf algebroids that also entails cotwisting or `quantisation' of the base and which is dual to a previous twisting construction of P. Xu. Whereas the latter applied the construction to the algebra of differential operators on a classical base $B$, we show that the dual of this is the algebra of sections $J(B)$ of the jet bundle and hence that the latter forms a Hopf algebroid. This is constructed for commutative algebras $B$ in a pro-object setting via quotients of the pair Hopf algebroid $B\otimes B$ and can then be deformed by our cotwist construction to give a possibly noncommutative jet Hopf algebroid over a noncommutative base. We also observe in the commutative case that $J^k(B)$ for jets of order $k$ can be identified with $J^1(B_k)$ where $B_k$ denotes $B$ equipped with a certain noncommutative first order differential calculus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02848 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum jet Hopf algebroids by cotwist Han, Xiao Majid, Shahn Quantum Algebra 16T05, 20G42, 46L87, 58B32 We introduce a cotwist construction for Hopf algebroids that also entails cotwisting or `quantisation' of the base and which is dual to a previous twisting construction of P. Xu. Whereas the latter applied the construction to the algebra of differential operators on a classical base $B$, we show that the dual of this is the algebra of sections $J(B)$ of the jet bundle and hence that the latter forms a Hopf algebroid. This is constructed for commutative algebras $B$ in a pro-object setting via quotients of the pair Hopf algebroid $B\otimes B$ and can then be deformed by our cotwist construction to give a possibly noncommutative jet Hopf algebroid over a noncommutative base. We also observe in the commutative case that $J^k(B)$ for jets of order $k$ can be identified with $J^1(B_k)$ where $B_k$ denotes $B$ equipped with a certain noncommutative first order differential calculus. |
| title | Quantum jet Hopf algebroids by cotwist |
| topic | Quantum Algebra 16T05, 20G42, 46L87, 58B32 |
| url | https://arxiv.org/abs/2507.02848 |