Torsion-Free Bimodule Connections and the Maximal Prolongation of a First-Order Differential Calculus
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915698576457728 |
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| author | Carotenuto, Alessandro Del Dono, Antonio Buachalla, Réamonn Ó Razzaq, Junaid |
| author_facet | Carotenuto, Alessandro Del Dono, Antonio Buachalla, Réamonn Ó Razzaq, Junaid |
| contents | We give an unexpectedly simple presentation of the maximal prolongation of a first-order differential calculus in terms of the bimodule map of a torsion-free bimodule connection. We then show that in the quantum homogeneous space case this simplifies even further. More explicitly, we show that the bimodule map associated to a bimodule connection, for any relative left Hopf module endowed with its canonical right module structure, admits a concise formula, given in terms of the adjont action of a Hopf algebra on a bimodule. %{\color{red} We also have the dual tangent space formula.} This is then used to derive sufficient conditions, in terms of the first-order differential forms, for the extendability of a first-order almost-complex structure. These results are applied to the quantum Grassmannian Heckenberger--Kolb calculi, yielding a simple uniform presentation of their degree two anti-holomorphic relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23579 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Torsion-Free Bimodule Connections and the Maximal Prolongation of a First-Order Differential Calculus Carotenuto, Alessandro Del Dono, Antonio Buachalla, Réamonn Ó Razzaq, Junaid Quantum Algebra We give an unexpectedly simple presentation of the maximal prolongation of a first-order differential calculus in terms of the bimodule map of a torsion-free bimodule connection. We then show that in the quantum homogeneous space case this simplifies even further. More explicitly, we show that the bimodule map associated to a bimodule connection, for any relative left Hopf module endowed with its canonical right module structure, admits a concise formula, given in terms of the adjont action of a Hopf algebra on a bimodule. %{\color{red} We also have the dual tangent space formula.} This is then used to derive sufficient conditions, in terms of the first-order differential forms, for the extendability of a first-order almost-complex structure. These results are applied to the quantum Grassmannian Heckenberger--Kolb calculi, yielding a simple uniform presentation of their degree two anti-holomorphic relations. |
| title | Torsion-Free Bimodule Connections and the Maximal Prolongation of a First-Order Differential Calculus |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2512.23579 |