Hopf Galois扩张和Frobenius扩张

Hopf Galois Extensions and Frobenius Extensions

设有限维Hopf代数H作用于代数A,A^H是A在这一作用下的不变子代数.在2中我们利用A#H与A^H之间的Morita context重新证明了A/A^H是右H-Golois扩张当且仅当它是H-Frobenius扩张且A作为左或右A＃H-模是忠实的.考虑H在A^k上导出的作用(?),其中K是H的正规子Hopf代数,(?)=H/K^+H,K^+=kerε(?)K,在3中我们证明当A/A^k是迹1扩张时,A/A^H的H-Frobenius扩张性质可以推出A^k/A^H((A^k)^(?)=A^H)的(?)-Frobenius扩张性质,因而利用2中所证定理,A/A^H的右H-Golois扩张性质可以推出A^k/a^H的右(?)-Galois扩张性质.

Assume that H is an algebra acted by a finite dimensional Hopf algebra H,with AHbeing the invariant subalgebra under H. In 2 , we use the Morita context between A # H and AH to prove anew that the extension A/AH is right H *-Galois if and onlg if it is H * -Frobe-nius and A as a left or right A#H-Module is faithful. We then consider the action H on AK induced by H,where K is a normal sub Hopfalgebra of H and H = H/K+H,K+ =kerε∩ K. We prove in 3 that if A/AKis a trace-one extension,then A/AHbeing H*-Frobenius implies AK/AH(AK)H=AH)being H*-Frobenius; thus by using the theorem proved in 2,A/ AH being right H*-Galois implies AK/AH being right H-Galois.