|Table of Contents|

[1] Gu Yue, Wang Wei, Wang Shuanhong,. Galois linear maps and their construction [J]. Journal of Southeast University (English Edition), 2019, 35 (4): 522-526. [doi:10.3969/j.issn.1003-7985.2019.04.016]
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Galois linear maps and their construction()
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Journal of Southeast University (English Edition)[ISSN:1003-7985/CN:32-1325/N]

Volumn:
35
Issue:
2019 4
Page:
522-526
Research Field:
Mathematics, Physics, Mechanics
Publishing date:
2019-12-30

Info

Title:
Galois linear maps and their construction
Author(s):
Gu Yue1;  Wang Wei2;  Wang Shuanhong1
1School of Mathematics, Southeast University, Nanjing 211189, China
2Nanjing Research Institute of Electronic Engineering, Nanjing 210007, China
Keywords:
Galois linear map;  antipode;  Hopf algebra;  Hopf(co)quasigroup
PACS:
O153.5
DOI:
10.3969/j.issn.1003-7985.2019.04.016
Abstract:
The condition of an algebra to be a Hopf algebra or a Hopf(co)quasigroup can be determined by the properties of Galois linear maps. For a bialgebra H, if it is unital and associative as an algebra and counital coassociative as a coalgebra, then the Galois linear maps T1 and T2 can be defined. For such a bialgebra H, it is a Hopf algebra if and only if T1 is bijective. Moreover, T-11 is a right H-module map and a left H-comodule map(similar to T2). On the other hand, for a unital algebra( no need to be associative), and a counital coassociative coalgebra A, if the coproduct and counit are both algebra morphisms, then the sufficient and necessary condition of A to be a Hopf quasigroup is that T1 is bijective, and T-11 is left compatible with ΔrrT-11 and right compatible with mlT-11 at the same time(The properties are similar to T2). Furthermore, as a corollary, the quasigroups case is also considered.

References:

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Memo

Memo:
Biographies: Gu Yue(1992—), female, graduate; Wang Shuanhong(corresponding author), male, doctor, professor, shuanhwang@seu.edu.cn.
Foundation items: The National Natural Science Foundation of China(No.11371088, 11571173, 11871144), the Natural Science Foundation of Jiangsu Province(No.BK20171348).
Citation: Gu Yue, Wang Wei, Wang Shuanhong. Galois linear maps and their construction[J].Journal of Southeast University(English Edition), 2019, 35(4):522-526.DOI:10.3969/j.issn.1003-7985.2019.04.016.
Last Update: 2019-12-20