Abstract
In this paper, we mainly focus on how to use Hom-partial actions to construct a new monoidal Hom–Hopf algebra. For this, we first introduce the notions of partial Hom–Smash products and partial Hom–Smash coproducts. Then, partial matched Hom-pairs are established to construct monoidal Hom–Hopf algebras, as application, some concrete examples are elaborated.
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References
Exel, R.: Circle actions on \(C^{*}\)-algebras, partial automorphisms and generalized Pimsner–Voiculescu exact sequences. J. Funct. Anal. 3(122), 361–401 (1994)
Exel, R.: Partial actions of groups and actions of inverse semigroups. Proc. Am. Math. Soc. 12(126), 3481–3494 (1998)
Dokuchaev, M., Exel, R., Piccione, P.: Partial representations and partial group algebras. J. Algebra 1(266), 251–268 (2000)
Dokuchaev, M., Exel, R.: Associativity of crossed products by partial actions, enveloping actions and partial representations. Trans. Am. Math. Soc. 5(357), 1931–1952 (2005)
Dokuchaev, M., Ferrero, M., Paques, A.: Partial actions and Galois theory. J. Pure Appl. Algebra 1(208), 77–87 (2000)
Caenepeel, S., Janssen, K.: Partial (co)actions of Hopf algebras and partial Hopf-Galois theory. Commun. Algebra 36, 2923–2946 (2008)
Alves, M.M.S., Batista, E.: Enveloping actions for partial Hopf actions. Commun. Algebra 38, 2872–2902 (2010)
Alves, M.M.S., Batista, E.: Partial Hopf actions, partial invariants and a Morita context. J. Algebra Discr. Math. 3, 1–19 (2009)
Alves, M.M.S., Batista, E.: Globalization theorems for partial Hopf (co)actions and some of their applications. Contemp. Math. 537, 13–30 (2011)
Hartwig, J., Larsson, D., Silvestrov, S.: Deformations of Lie algebras using \(\sigma \)-derivations. J. Algebra 295, 314–361 (2006)
Makhlouf, A., Silvestrov, S.: Hom-algebra structures. J. Gen. Lie Theory Appl. 2, 51–64 (2008)
Makhlouf, A., Silvestrov, S.D.: Hom–Lie admissible Hom-coalgebra, Hom–Hopf algebras. In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (eds.) Generalzied Lie Theory in Mathematics, Physics and Beyond, pp. 189–206. Springer, Berlin (2009)
Makhlouf, A., Silvestrov, S.D.: Hom-algebras and Hom-coalgebras. J. Pure Appl. Algebra 9, 553–589 (2010)
Yau, D.: Hom-bialgebras and comodule Hom-algebras. Int. Electron. J. Algebra 8, 45–64 (2010)
Caenepeel, S., Goyvaerts, I.: Monoidal Hom–Hopf algebras. Commun. Algebra 39, 2216–2240 (2011)
Singer, W.: Extension theory for connected Hopf algebras. J. Algebra 21, 1–16 (1972)
Takeuchi, M.: Matched pairs of groups and bismash products of Hopf algebras. Commun. Algebra 9, 841–882 (1981)
Azevedo, D., Martini, G., paques, A., Silva, L.: Hopf algebras arising from partial (co)actions. J. Algebra Appl. 1(20), 2140006 (2021)
Makhlouf, A., Panaite, F.: Hom-L-R-smash products, Hom-diagonal crossed products and the Drinfeld double of a Hom-Hopf algebra. J. Algebra 441, 314–343 (2015)
Makhlouf, A., Panaite, F.: Twisting operators, twisted tensor products and smash prodcuts for Hom-associative algebras. Glasgow Math. J. 58, 513–538 (2016)
Chen, Q.-G., Cheng, W.-J.: Twisted partial actions of monoidal Hom–Hopf algebras. Filomat 9(36), 2991–3011 (2022)
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Communicated by Rasool Hafezi.
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Jia, L. The Monoidal Hom–Hopf Algebra Arising From Partial Hom-Actions. Bull. Iran. Math. Soc. 50, 41 (2024). https://doi.org/10.1007/s41980-024-00873-0
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DOI: https://doi.org/10.1007/s41980-024-00873-0