THE HOM-TWISTED SMASH PRODUCT BIALGEBRAS

Jiafeng Lü; Wenying Yu; Ling Liu

doi:10.11948/20200477

2021 Volume 11 Issue 3

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Jiafeng Lü, Wenying Yu, Ling Liu. THE HOM-TWISTED SMASH PRODUCT BIALGEBRAS[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1652-1662. doi: 10.11948/20200477

Citation:

Jiafeng Lü, Wenying Yu, Ling Liu. THE HOM-TWISTED SMASH PRODUCT BIALGEBRAS[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1652-1662. doi: 10.11948/20200477

THE HOM-TWISTED SMASH PRODUCT BIALGEBRAS

1.
Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

Corresponding author: Email: ntliulin@zjnu.cn(L. Liu)
Fund Project: The authors were supported by National Natural Science Foundation of China (Nos. 11801515, 12071441), the Natural Science Foundation of Zhejiang Province (No. LY20A010003) and the Foundation of Zhejiang Educational Committee (No. Y201942625)

Abstract

Abstract

Let $ (H, {\alpha}_{H}) $ be a Hom-Hopf algebra and $ (A, \alpha_{A}) $ be an $ (H, {\alpha}_{H}) $-Hom-bimodule algebra with the maps $ {\alpha}_A, {\alpha}_H $ bijective. Then in this paper, we first introduce the notion of Hom-twisted smash product $ (A{\star} H, {\alpha}_{A}{\star} {\alpha}_{H}) $ and then study the conditions for the Hom-twisted smash product and tensor coproduct to form a Hom-bialgebra and a Hom-Hopf algebra. Furthermore, we give a non-trival example of Hom-twisted smash product Hopf algebra and a characterization of left $ (A{\star} H, {\alpha}_{A}{\star} {\alpha}_{H}) $-Hom module.
- Hom-bialgebra /
- Hom-twisted smash product /
- tensor coproduct
MSC: 16T05

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References

[1]	V. Abramov and S. Silvestrov, 3-Hom-Lie algebras based on $\sigma $-derivation and involution, Adv. Appl. Clifford Algebra, 2020, 30(3), 45, 13 pp. doi: 10.1007/s00006-020-01068-6 CrossRef Google Scholar
[2]	N. Aizawa and H. Sato, $q$-deformation of the Virasoro algebra with central extension, Phys. Lett. B, 1991, 256(2), 185-190. doi: 10.1016/0370-2693(91)90671-C CrossRef Google Scholar
[3]	S. Caenepeel and I. Goyvaerts, Monoidal Hom-Hopf algebras, Comm. Algebra, 2011, 39(6), 2216-2240. doi: 10.1080/00927872.2010.490800 CrossRef Google Scholar
[4]	Y. Chen, H. Zheng and L. Zhang, Double Hom-associative algebra and double Hom-Lie bialgebra, Adv. Appl. Clifford Algebra, 2020, 30(1), 8, 25 pp. doi: 10.1007/s00006-019-1028-2 CrossRef Google Scholar
[5]	Y. Doiand M. Takeuchi, Multiplication alteration by two-cocycles in the quantum view, Comm. Algebra, 1994, 22(14), 5715-5732. doi: 10.1080/00927879408825158 CrossRef Google Scholar
[6]	V. G. Drinfeld, Quantum groups, Proceedings of the International Congress of Mathematicians, Berkeley, 1986. Google Scholar
[7]	N. Hu, $q$-Witt algebras, $q$-Lie algebras, $q$-holomorph structure and representations, Algebra Colloq., 1999, 6(1), 51-70. Google Scholar
[8]	L. Liu and Q. L. Guo, On Twisted Smash Products of Monoidal Hom-Hopf Algebras, Comm. Algebra, 2016, 44(10), 4140-4164. doi: 10.1080/00927872.2015.1087003 CrossRef Google Scholar
[9]	L. Liu, A. Makhlouf, C. Menini and F. Panaite, BiHom-Novikov algebras and infinitesimal BiHom-bialgebras, J. Algebra, 2020, 560, 1146-1172. doi: 10.1016/j.jalgebra.2020.06.012 CrossRef Google Scholar
[10]	T. Ma, H. Yang, L. Liu and Q. Chen, On unified Hom-Yetter-Drinfeld categories, J. Geom. Phys., 2019, 144(7), 81-107. Google Scholar
[11]	A. Makhlouf and S. Silvestrov, Hom-algebras structures, J. Gen. Lie Theory Appl., 2008, 2(2), 52-64. Google Scholar
[12]	A. Makhlouf and S. D. Silvestrov, Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras, in Generalized Lie Theory in Mathematics, Physics and Beyond, Springer Verlag, Berlin, 2009. Google Scholar
[13]	A. Makhlouf and F. Panaite, Hom-L-R-smash products, Hom-diagonal crossed products and the Drinfel'd double of a Hom-Hopf algebra, J. Algebra, 2015, 441, 341-343. Google Scholar
[14]	R. Molnar, Semi-direct products of Hopf algebras, J. Algebra, 1977, 47(1), 29-51. doi: 10.1016/0021-8693(77)90208-3 CrossRef Google Scholar
[15]	S. Wang and J. Li, On twisted smash products for bimodule algebras and the Drinfeld double, Comm. Algebra, 1998, 26(8), 2435-2444. doi: 10.1080/00927879808826288 CrossRef Google Scholar
[16]	D. Yau, Hom-quantum groups Ⅰ: Quasi-triangular Hom-bialgebras, J. Phys. A: Math. Theor., 2012, 45(6), 065203. doi: 10.1088/1751-8113/45/6/065203 CrossRef Google Scholar