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Two results on double crossed biproducts

Year 2022, Volume: 51 Issue: 4, 1121 - 1140, 01.08.2022
https://doi.org/10.15672/hujms.997154

Abstract

Let HH be an algebra with a distinguished element εHHεH∈H∗ and C,DC,D two coalgebras. Based on the construction of Brzeziński’s crossed coproduct, under some suitable conditions, we introduce a coassociative coalgebra C×GTHβR×DC×TGHRβ×D which is a more general two-sided coproduct structure including two-sided smash coproduct. Necessary and sufficient conditions for C×GTHβR×DC×TGHRβ×D equipped with two-sided tensor product algebra CHDC⊗H⊗D to be a bialgebra (Hopf algebra) are provided. On the other hand, we obtain an improved version of the double crossed biproduct CαHβDC⋆αHβ⋆D in [An extended form of Majid's double biproduct, J. Algebra Appl. 16 (4), 1760061, 2017] which induces a description of CαHβDC⋆αHβ⋆D similar to Majid double biproduct CHDC⋆H⋆D and also present some related structures.

References

  • [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.
Year 2022, Volume: 51 Issue: 4, 1121 - 1140, 01.08.2022
https://doi.org/10.15672/hujms.997154

Abstract

References

  • [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.
There are 14 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Tianshui Ma 0000-0003-1275-7214

Bei Li This is me 0000-0002-2449-7387

Publication Date August 1, 2022
Published in Issue Year 2022 Volume: 51 Issue: 4

Cite

APA Ma, T., & Li, B. (2022). Two results on double crossed biproducts. Hacettepe Journal of Mathematics and Statistics, 51(4), 1121-1140. https://doi.org/10.15672/hujms.997154
AMA Ma T, Li B. Two results on double crossed biproducts. Hacettepe Journal of Mathematics and Statistics. August 2022;51(4):1121-1140. doi:10.15672/hujms.997154
Chicago Ma, Tianshui, and Bei Li. “Two Results on Double Crossed Biproducts”. Hacettepe Journal of Mathematics and Statistics 51, no. 4 (August 2022): 1121-40. https://doi.org/10.15672/hujms.997154.
EndNote Ma T, Li B (August 1, 2022) Two results on double crossed biproducts. Hacettepe Journal of Mathematics and Statistics 51 4 1121–1140.
IEEE T. Ma and B. Li, “Two results on double crossed biproducts”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 4, pp. 1121–1140, 2022, doi: 10.15672/hujms.997154.
ISNAD Ma, Tianshui - Li, Bei. “Two Results on Double Crossed Biproducts”. Hacettepe Journal of Mathematics and Statistics 51/4 (August 2022), 1121-1140. https://doi.org/10.15672/hujms.997154.
JAMA Ma T, Li B. Two results on double crossed biproducts. Hacettepe Journal of Mathematics and Statistics. 2022;51:1121–1140.
MLA Ma, Tianshui and Bei Li. “Two Results on Double Crossed Biproducts”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 4, 2022, pp. 1121-40, doi:10.15672/hujms.997154.
Vancouver Ma T, Li B. Two results on double crossed biproducts. Hacettepe Journal of Mathematics and Statistics. 2022;51(4):1121-40.