Two results on double crossed biproducts

Yıl 2022, Cilt: 51 Sayı: 4, 1121 - 1140, 01.08.2022

Tianshui Ma Bei Li

https://doi.org/10.15672/hujms.997154

Öz

Let $H$ be an algebra with a distinguished element ${\epsilon }_H\in H^{\ast }$ and $C,D$ two coalgebras. Based on the construction of Brzeziński’s crossed coproduct, under some suitable conditions, we introduce a coassociative coalgebra $C\times {}_T^G{H}_R^{\beta }\times D$ which is a more general two-sided coproduct structure including two-sided smash coproduct. Necessary and sufficient conditions for $C\times {}_T^G{H}_R^{\beta }\times D$ equipped with two-sided tensor product algebra $C\otimes H\otimes D$ to be a bialgebra (Hopf algebra) are provided. On the other hand, we obtain an improved version of the double crossed biproduct $C\star {}^{\alpha }{H}^{\beta }\star D$ in [An extended form of Majid's double biproduct, J. Algebra Appl. 16 (4), 1760061, 2017] which induces a description of $C\star {}^{\alpha }{H}^{\beta }\star D$ similar to Majid double biproduct $C\star H\star D$ and also present some related structures.

Anahtar Kelimeler

crossed coproduct, Radford biproduct, double biproduct

Kaynakça

• [1] T. Brzeziński, Crossed products by a coalgebra, Comm. Algebra, 25, 3551-3575, 1997.
• [2] S. Caenepeel, D.G. Wang and Y.X. Wang, Twistings, crossed coproducts, and Hopf-Galois coextensions, Int. J. Math. Math. Sci. 69, 4325-4345, 2003.
• [3] J.J. Guo and W.Z. Zhao, Bialgebra structure on crossed coproducts in Yetter-Drinfeld category, Southeast Asian Bull. Math. 34, 663-684, 2010.
• [4] B. I-P. Lin, Crossed coproducts of Hopf algebras, Comm. Algebra, 10, 1-17, 1982.
• [5] T.S. Ma, H.Y. Li and S.X. Xu, Construction of a braided monoidal category for Brzeziński crossed coproducts of Hopf $\pi$-algebras, Colloq. Math. 149 (2), 309-323, 2017.
• [6] T.S. Ma and L.L. Liu, Rota-Baxter coalgebras and Rota-Baxter bialgebras, Linear Multilinear Algebra, 64 (5), 968-979, 2016.
• [7] T.S. Ma and S.H. Wang, Bitwistor and quasitriangular structures of bialgebras, Comm. Alge- bra, 38 (9), 3206-3242, 2010.
• [8] T.S. Ma and H.H. Zheng, An extended form of Majid’s double biproduct, J. Algebra Appl. 16 (4), 1750061, 2017.
• [9] S. Majid, New quantum groups by double-bosonisation, Czechoslovak J. Phys. 47 (1), 79-90, 1997.
• [10] S. Majid, Double-bosonization of braided groups and the construction of $U_q(g)$, Math. Proc. Cambridge Philos. Soc. 125 (1), 151-192, 1999.
• [11] F. Panaite, Equivalence of crossed coproducts, Bull. Belg. Math. Soc. Simon Stevin, 6, 259- 278, 1999.
• [12] D.E. Radford, Hopf Algebras, KE Series on Knots and Everything, Vol. 49, World Scientic, New Jersey, 2012.
• [13] G.D. Shi and S.H. Wang, Schur-Weyl quasi-duality and (co)triangular Hopf quasigroups, J. Math. Phys. 61, 051701, 2020.
• [14] S.H. Wang, D.G. Wang and Z.P. Yao, Hopf algebra structure over crossed coproducts, South- east Asian Bull. Math. 24 (1), 105-113, 2000.