Abstract
Let k be a field, H a Hopf algebra with a bijective antipode, and A an H-dimodule algebra. We assume that there is an H-colinear algebra map from H to A. We generalize the Fundamental Theorem of (A, H)-Hopf modules to (A, H)-dimodules. For example, A could be a coquasitriangular Hopf algebra. When H is the group algebra kG of a multiplicatively abelian group G, an (A, kG)-dimodule is just a G-graded A-module which is an \(A\#G\)-module and a G-dimodule. We also prove the Fundamental Theorem for Hopf Yetter–Drinfeld (A, H)-modules, when H is cocommutative and A is a Hopf Yetter–Drinfeld H-module algebra. For example, A could be the regular Hopf Yetter–Drinfeld H-module algebra H or \(H^{op}\).
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References
Bagheri, S., Wisbauer, R.: Hom-tensor relations for two-sided Hopf modules over quasi-Hopf algebras. Commun. Algebra 40(9), 3257–3287 (2012)
Caenepeel, S., Beattie, M.: A cohomological approach to the Brauer-Long group and the groups of Galois extensions and strongly graded rings. Trans. Am. Math. Soc. 324(2), 747–775 (1991)
Caenepeel, S., Van Oystaeyen, F., Zhang, Y.H.: Quantum Yang-Baxter module algebras. K- Theory 8(3), 231–255 (1994)
Caenepeel, S., Van Oystaeyen, F., Zhang, Y.H.: The Brauer group of Yetter–Drinfeld module algebras. Trans. Am. Math. Soc. 349(9), 3737–3771 (1991)
Cohen, M., Fischman, D., Montgomery, S.: On Yetter–Drinfeld categories and H-commutativity, Commun. Algebra 27(3), 1321–1345 (1999)
Doi, Y.: On the structure of relative Hopf modules. Commun. Algebra 11(3), 243–255 (1983)
Guédénon, T., Nango, C.L.: A Brauer–Clifford–Long group for the category of dyslectic \((S, H)\)-dimodule algebras. São Paulo J. Math. Sciences 16(2), 803–848 (2022). https://doi.org/10.1007/s40863-021-00242-3
Hausser, F., Nill, F.: Integral theory for quasi-Hopf algebras, Preprint, (Math. QA 13 May 1999). arXiv:math/9904164v2
Long, F.W.: The Brauer group of dimodule algebra. J. Algebra 31(3), 559–601 (1974)
Montgomery, S.: Hopf algebras and their actions on rings. CBMS Reg. Conf. Ser. in Math. 82, AMS, Providence RI, (1993)
Sweedler, M.: Hopf algebras. Benjamin, New York (1969)
Yin, L.: Hopf modules in the category of Yetter–Drinfeld modules. Appl. Math. 7, 629–637 (2016)
Zhang, L.-Y., Tong, W.-T.: Quantum Yang-Baxter \(H\)-module algebras and their braided products. Commun. Algebra 31(5), 2471–2495 (2003)
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Guédénon, T. Fundamental Theorem for (A, H)-dimodules. Beitr Algebra Geom (2023). https://doi.org/10.1007/s13366-023-00726-7
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DOI: https://doi.org/10.1007/s13366-023-00726-7
Keywords
- Hopf algebra
- H-module algebra
- H-comodule algebra
- H-dimodule
- H-dimodule algebra
- \((A, H)\)-dimodule
- Hopf Yetter–Drinfeld H-module
- Hopf Yetter–Drinfeld H-module algebra
- Hopf Yetter–Drinfeld \((A, H)\)-module