Galois linear maps and their construction

Galois线性映射及其构造（英文）

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,T1^-1 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 T1^-1 is left compatible with ΔT1-11^r and right compatible with mT1-1^l 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,T1-1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T1-1与右余积映射ΔT1-1r左相容,同时与左积映射mT1-1l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.