Hom-coalgebra cleft extensions and braided tensor Hom-categories of Hom-entwining structures

Year 2019, Volume: 48 Issue: 1, 186 - 199, 01.02.2019

Chen Quanguo Wang Dingguo

Abstract

We investigate how the category of Hom-entwined modules can be made into a monoidal category. The sufficient and necessary conditions making the category of Hom-entwined modules have a braiding are given. Also, we formulate the concept of Hom-cleft extension for a Hom-entwining structure, and prove that if $(A, \alpha)$ is a $(C,\gamma)$-cleft extension, then there is an isomorphism of Hom-algebras between $(A, \alpha)$ and  a crossed product Hom-algebra of $A^{coC}$ and $C$.

Keywords

Hom-Hopf algebra, Hom-entwining structure, cleft extension

References

• J.N. Alonso Álvarez, J.M. Fernández Vilaboaa, R. González Rodriguez, et al. Weak C-cleft extensions, weak entwining structures and weak Hopf algebras, J. Algebra 284, 679–704, 2005.
• T. Brzezinski and S. Majid, Coalgebra bundles, Comm. Math. Phys. 191, 467–492, 1998.
• T. Brzezinski, On modules associated to coalgebra Galois extensions, J. Algebra 215, 290–317, 1999.
• S. Caenepeel and I. Goyvaerts Monoidal Hom-Hopf algebras, Comm. Algebra 39 (6), 2216–2240, 2011.
• S. Caenepeel, F. van Oystaeyen and B. Zhou, Making the category of Doi-Hopf modules into a braided monoidal category, Algebr. Represent. Theory 1 (1), 75–96, 1998.
• Q.G. Chen and D.G. Wang, Constructing quasitriangular Hopf algebras, Comm. Algebra 43 (4), 1698–1722, 2015.
• Q.G. Chen and D.G. Wang, A class of coquasitriangular Hopf group algebras, Comm. Algebra 44 (1), 310–335, 2016.
• Q.G. Chen and D.G. Wang, A duality Theorem for L-R crossed product, Filomat 30 (5), 1305–1313, 2016.
• Q.G. Chen, D.G. Wang and X.D. Kang, Twisted partial coactions of Hopf algebras, Front. Math. China 12, 63–86, 2017.
• Y.Y. Chen, Z.W. Wang and L.Y. Zhang, The fundamental theorem and Maschke’s theorem in the category of relative Hom-Hopf modules, Colloq. Math. 144 (1), 55–71, 2016.
• Y. Doi and M. Takeuchi, Cleft comodule algebras for a bialgebra, Comm. Algebra 14, 801–817, 1986.
• J.M. Fernández Vilaboa and E. Villanueva Novoa, A Characterization of the cleft comodule triples, Comm. Algebra 16, 613–622, 1988.
• S.J. Guo, X.H. Zhang and S.X. Wang, Braided monoidal categories and Doi-Hopf modules for monoidal Hom-Hopf algebras, Colloq. Math. 143 (1), 79–103, 2016.
• S. Karacuha, Hom-entwining structures and Hom-Hopf-type modules, arXiv:1412.2002v2.
• H.F. Kreimer and M. Takeuchi, Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J. 30, 675–691, 1981.
• L. Liu and B.L. Shen, Radford’s biproducts and Yetter-Drinfel’d modules for monoidal Hom-Hopf algebras, J. Math. Phys. 55, 031701, 2014.
• L. Liu and S.H. Wang, Making the category of entwined modules into a braided monoidal category, J. Southeast Univ. (English Ed.) 24 (2), 250–252, 2008.
• A. Makhlouf and F. Panaite, Yetter-Drinfeld modules for Hom-bialgebras, J. Math. Phys. 55, 013501, 2014.
• A. Makhlouf and S.D. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2(2), 51–64, 2008.
• A. Makhlouf and S.D. Silvestrov, Hom-algebras and Hom-coalgebras, J. Algebra Appl. 9 (4), 553–589, 2010.
• D. Yau, Hom-Yang-Baxter-equation, Hom-Lie algebras and quasitriangular bialgebras, J. Phys. A 42, 165202, 2009.
• D. Yau, Hom-algebras and homology, J. Lie Theory 19, 409–421, 2009.
• D. Yau, Hom-quantum groups I: Quasitriangular Hom-bialgebras, J. Phys A 45, 065203, 2012.
• X.F. Zhao and X.H. Zhang, Lazy 2-cocycles over monoidal Hom-Hopf algebras, Colloq. Math. 142, 61–81, 2016.