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[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()
Galois线性映射及其构造
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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
Galois线性映射及其构造
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
谷乐1;  王伟2;  王栓宏1
1东南大学数学学院, 南京 211189; 2中国电子科技集团公司第28研究所, 南京 210007
Keywords:
Galois linear map;  antipode;  Hopf algebra;  Hopf(co)quasigroup
Galois线性映射;  对极;  Hopf代数;  Hopf(余)拟群
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.
一个代数构成Hopf代数或Hopf(余)拟群的条件可由Galois线性映射的性质来确定.对于一个双代数H, 如果其作为代数是结合有单位的, 且作为余代数是余结合有余单位的, 则可以定义Galois线性映射T1和T2.对于一个结合余结合的双代数H(有单位和余单位), 则H为一个Hopf代数当且仅当Galois线性映射T1是双射, 且进一步地, T-11是右H-模和右H-余模映射.另一方面, 对于一个有单位的代数A(不一定是结合的), A作为余代数是余结合有余单位的, 如果A的余乘法和余单位均为代数同态, 则A为一个Hopf拟群当且仅当 Galois线性映射T1是双射且T-11与右余积映射ΔrT<sup>-1</sub>11左相容, 同时与左积映射mlT<sup>-1</sub>11右相容(相似的性质也适用于Galois线性映射T2).作为推论, 拟群的情形也得到了讨论.11右相容(相似的性质也适用于Galois线性映射).作为推论, 拟群的情形也得到了讨论.11左相容, 同时与左积映射11右相容(相似的性质也适用于Galois线性映射).作为推论, 拟群的情形也得到了讨论.

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