Double biproduct Hom-bialgebra and related quasitriangular structures

Abstract

Let (H, β) be a Hom-bialgebra such that β ² = id _H. (A, α _A) is a Hom-bialgebra in the left-left Hom-Yetter-Drinfeld category _H ^HYD and (B, α _B) is a Hom-bialgebra in the right-right Hom-Yetter-Drinfeld category YD_H ^H. The authors define the two-sided smash product Hom-algebra $\left( {A\natural H\natural B,{\alpha _A} \otimes \beta \otimes {\alpha _B}} \right)$ and the two-sided smash coproduct Homcoalgebra $\left( {A\diamondsuit H\diamondsuit B,{\alpha _A} \otimes \beta \otimes {\alpha _B}} \right)$. Then the necessary and sufficient conditions for $\left( {A\natural H\natural B,{\alpha _A} \otimes \beta \otimes {\alpha _B}} \right)$ and $\left( {A\diamondsuit H\diamondsuit B,{\alpha _A} \otimes \beta \otimes {\alpha _B}} \right)$ to be a Hom-bialgebra (called the double biproduct Hom-bialgebra and denoted by $\left( {A_\diamondsuit ^\natural H_\diamondsuit ^\natural B,{\alpha _A} \otimes \beta \otimes {\alpha _B})} \right)$ are derived. On the other hand, the necessary and sufficient conditions for the smash coproduct Hom-Hopf algebra $\left( {A\diamondsuit H,{\alpha _A} \otimes \beta } \right)$ to be quasitriangular are given.

Keywords

Double biproduct / Hom-Yetter-Drinfeld category / Radford’s biproduct / Hom-Yang-Baxter equation

References

[1] Andruskiewitsch N., Schneider H.-J.. On the classification of finite-dimensional pointed Hopf algebras. Ann. Math., 2010, 171(1): 375-417

[2] Caenepeel S., Goyvaerts I.. Monoidal Hom-Hopf algebras. Comm. Algebra, 2011, 39(6): 2216-2240

[3] Caenepeel S., Ion B., Militaru G., Zhu S. L.. The factorization problem and the smash biproduct of algebras and coalgebras. Algebra Represent. Theory, 2000, 3: 19-42

[4] Hu N.. q-Witt algebras, q-Lie algebras, q-holomorph structure and representations. Algebra Colloq., 1999, 6(1): 51-70

[5] Kassel C.. Quantum Groups, Graduate Texts in Mathematics, 1995, Berlin: Springer-Verlag

[6] Li H. Y., Ma T. S.. A construction of Hom-Yetter-Drinfeld category. Colloq. Math., 2014, 137(1): 43-65

[7] Ma T. S., Jiao Z. M., Song Y. N.. On crossed double biproduct. J. Algebra Appl., 2013, 12(5): 1250211

[8] Ma T. S., Li H. Y.. On Radford biproduct. Comm. Algebra, 2015, 43(9): 3946-3966

[9] Ma T. S., Li H. Y., Yang T.. Cobraided smash product Hom-Hopf algebras. Colloq. Math., 2014, 134(1): 75-92

[10] Ma T. S., Wang S. H.. Bitwistor and quasitriangular structures of bialgebras. Comm. Algebra, 2010, 38(9): 3206-3242

[11] Majid S.. Double-bosonization of braided groups and the construction of Uq(g). Math. Proc. Cambridge Philos. Soc., 1999, 125(1): 151-192

[12] Makhlouf A., Panaite F.. Yetter-Drinfeld modules for Hom-bialgebras. J. Math. Phys., 2014, 55: 013501

[13] Makhlouf A., Panaite F.. Twisting operators. twisted tensor products and smash products for Homassociative algebras, Glasgow Math. J., 2016, 58: 513-538

[14] Makhlouf A., Silvestrov S. D.. Hom-algebra stuctures. J. Gen. Lie Theory Appl., 2008, 2: 51-64

[15] Makhlouf A., Silvestrov S. D.. Hom-algebras and hom-coalgebras. J. Algebra Appl., 2010, 9: 553-589

[16] Molnar R. K.. Semi-direct products of Hopf algebras. J. Algebra, 1977, 47: 29-51

[17] Radford D. E.. The structure of Hopf algebra with a projection. J. Algebra, 1985, 92: 322-347

[18] Radford D. E.. Biproducts and Kashina’s examples. Comm. Algebra, 2014, 44(1): 174-204

[19] Sweedler M. E.. Hopf Algebras, 1969, New York: Benjamin

[20] Taft E. J.. The order of the antipode of finite dimensional Hopf algebra. Proc. Nat. Acad. Sci. USA, 1971, 68: 2631-2633

[21] Yau, D., Module Hom-algebras. arXiv:0812.4695v1

[22] Yau D.. Hom-bialgebras and comodule Hom-algebras. Int. Electron. J. Algebra, 2010, 8: 45-64

[23] Yau D.. Hom-quantum groups I, quasitriangular Hom-bialgebras. J. Phys. A, 2012, 45(6): 065203

[24] Yau, D., Hom-quantum groups II,cobraided Hom-bialgebras and Hom-quantum geometry. arXiv:0907.1880

[25] Yau, D., Hom-quantum groups III, representations and module Hom-algebras. arXiv:0911.5402

[26] Yau D.. The Hom-Yang-Baxter equation and Hom-Lie algebras. J. Math. Phys., 2011, 52: 053502

[27] Zhang T., Li J.. Comodule Hom-coalgebras. Int. J. Algebra and Statistics, 2013, 2(1): 57-63