Fundamental theorem of \((A,\mathcal G,H)\)-comodules
Let \(k\) be a field, \(H\) a Hopf algebra with a bijective antipode, \(\mathcal G\) an \(H\)-comodule Lie algebra and \(A\) a commutative \(({\mathcal G},H)\)-comodule algebra. We assume that there is an \(H\)-colinear algebra map from \(H\) to \(A^{\mathcal G}\). We generalize the Fundamental Theo...
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| Date: | 2025 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2025
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2345 |
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| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | Let \(k\) be a field, \(H\) a Hopf algebra with a bijective antipode, \(\mathcal G\) an \(H\)-comodule Lie algebra and \(A\) a commutative \(({\mathcal G},H)\)-comodule algebra. We assume that there is an \(H\)-colinear algebra map from \(H\) to \(A^{\mathcal G}\). We generalize the Fundamental Theorem of \((A,H)\)-Hopf modules to \((A,{\mathcal G},H)\)-comodules, and we deduce relative projectivity in the category of \((A,{\mathcal G},H)\)-comodules. In many applications, \(A\) could be a commutative \(G\)-graded \(\mathcal G\)-module algebra, where \(G\) is an abelian group and \(\mathcal G\) is a \(G\)-graded Lie algebra; or a rational \(({\mathcal G},G)\)-module algebra, where \(G\) is an affine algebraic group and \(\mathcal G\) is a rational \(G\)-module Lie algebra. |
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