[1] Zhang Tao, Wang Shuanhong,. Sweedler’s dual of Hopf algebras in HHYDQCM^- [J]. Journal of Southeast University (English Edition), 2020, 36 (3): 364-366. [doi:10.3969/j.issn.1003-7985.2020.03.016]

Copy

Sweedler’s dual of Hopf algebras in ^H_HYDQCM^-()

Share：

Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:

36

Issue:

2020 03

Page:

364-366

Research Field:

Mathematics, Physics, Mechanics

Publishing date:

2020-09-20

Info

Title:

Sweedler’s dual of Hopf algebras in ^H_HYDQCM^-

Author(s):

Zhang Tao;  Wang Shuanhong

School of Mathematics, Southeast University, Nanjing 211189, China

Keywords:

Hopf(co)quasigroup;  Yetter-Drinfeld quasi(co)module;  braided monoidal category;  duality

PACS:

O153.5

DOI:

10.3969/j.issn.1003-7985.2020.03.016

Abstract:

Firstly, the notion of the left-left Yetter-Drinfeld quasicomodule M=(M, ·, ρ)over a Hopf coquasigroup H is given, which generalizes the left-left Yetter-Drinfeld module over Hopf algebras. Secondly, the braided monoidal category ^H_HYDQCM^- is introduced and the specific structure maps are given. Thirdly, Sweedler’s dual of infinite-dimensional Hopf algebras in ^H_HYDQCM^- is discussed. It proves that if(B, m_B, μ_B, Δ_B, ε_B)is a Hopf algebra in ^H_HHYDQCM^- with antipode S_B, then(B⁰, (m_B⁰)^op, ε^*_B, (Δ_B⁰)^op, μ^*_B)is a Hopf algebra in ^H_HYDQCM^- with antipode S^*_B, which generalizes the corresponding results over Hopf algebras.

References:

[1] Schauenburg P. Hopf modules and Yetter-Drinfel’d modules[J]. Journal of Algebra, 1994, 169: 874–890.
[2] Caenepeel S, Militaru G, Zhu S L. Frobenius and separable functors for entwined modules[C]//Lecture Notes in Mathematics. Heidelberg: Springer, 2002: 89-157.
[3] Wang S H. Braided monoidal categories associated to yetter-drinfeld categories[J]. Communications in Algebra, 2002, 30(11): 5111-5124.
[4] Doi Y. Hopf modules in Yetter-Drinfeld categories[J]. Communications in Algebra, 1998, 26(9): 3057-3070.
[5] Klim J, Majid S. Hopf quasigroups and the algebraic 7-sphere[J]. Journal of Algebra, 2010, 323(11): 3067-3110.
[6] Alonso Álvarez J N, Fernández Vilaboa J M, González Rodríguez R, et al. Projections and Yetter-Drinfel’d modules over Hopf(co)quasigroups[J]. Journal of Algebra, 2015, 443: 153-199.
[7] Brzeziński T. Hopf modules and the fundamental theorem for Hopf(co)quasigroups[J]. International Electron Journal of Algebra, 2010, 8:114-128.
[8] Brzeziński T, Jiao Z M. Actions of Hopf quasigroups[J]. Communications in Algebra, 2012, 40(2): 681-696.
[9] Fang X L, Wang S H. Twisted smash product for Hopf quasigroups [J]. Journal of Southeast University(English Edition), 2011, 27(3): 343-346.
[10] Zhang T, Wang S H, Wang D G. A new approach to braided monoidal categories [J]. Journal of Mathematical Physics, 2019, 60(1): 013510. DOI:10.1063/1.5055707.
[11] Pérez-Izquierdo J M. Algebras, hyperalgebras, nonassociative bialgebras and loops[J]. Advances in Mathematics, 2007, 208(2): 834-876.
[12] Sweedler M E. Hopf algebras[M]. New York: W A Benjamin Inc, 1969.
[13] Ng S H, Taft E J. Quantum convolution of linearly recursive sequences[J]. Journal of Algebra, 1997, 198(1): 101-119.

Memo

Memo:

Biographies: Zhang Tao(1990—), male, Ph.D. candidate; Wang Shuanhong(corresponding author), male, doctor, professor, Shuanwang@seu.edu.cn.
Foundation item: The National Natural Science Foundation of China(No.11371088, 11571173, 11871144).
Citation: Zhang Tao, Wang Shuanhong. Sweedler’s dual of Hopf algebras in ^H_HYDQCM^-.[J].Journal of Southeast University(English Edition), 2020, 36(3):364-366.DOI:10.3969/j.issn.1003-7985.2020.03.016.

Common functions