H-交换代数扭曲冲积的Maschke型定理  

A Maschke Type Theorem for Twisted Smash Products of H-commutative Algebras

作　　者：郭双建[1] 董丽红[2] 

机构地区：[1]贵州财经大学数学与统计学院,贵阳550025 [2]河南师范大学数学与信息科学学院,河南新乡453007

出　　处：《重庆师范大学学报（自然科学版）》2016年第1期55-57,共3页Journal of Chongqing Normal University:Natural Science

基　　金：国家自然科学基金青年基金(No.11101128);国家自然科学基金数学天元基金(No.11426073);江苏省自然科学基金(No.BK2012736);贵州省科学技术基金(No.2014GZ81365);贵州省高校优秀科技创新人才支持计划(No.黔教合KY字[2015]481);贵州省教育厅"125"重大科技专项资助项目(No.黔教合重大专项字[2012]011)

摘　　要：利用著名的Maschke型定理讨论了H-交换代数扭曲冲积的半单性,设H为域k上的有限维Hopf代数带有非退化的积分t,A是Yetter-Drinfeld模代数和H-双模代数,并且是H交换代数,根据已有文献的工作,给出了H交换代数扭曲冲积的Maschke型定理,通过对H中的积分和H的投射性质的研究,刻画了扭曲冲积A#H的半单性。利用Hopf代数的模论和双模代数的性质,对任意的左A#H-模M和N,定义了H的右A-模结构,并且验证了H是A-A双模,并讨论了A#H-模范畴中的态射集的性质与其上的模作用,证明了HomA(M,N)是一个左A#H-模,从而得到(HomA(M,N))H=A#HHom(A,M),并且进一步研究了A的投射性质。若假设A是半单的,得到A#H是半单的当且仅当A是投射的左A#H-模。最后给出在A是半单的前提下,则A#H是半单的当且仅当t·c=1对某个C∈A。The paper is discussed with the semisimplicity of twisted smash products of H-commutative algebras by famous Maschke type theorem. Let H be a finite dimensional Hopf algebra over a field k with a non-degenerate integral t and A Yetter-Drinfeld module algebra and H-bimodule algebra, and H-commutative algebra. Based on the work of Wang and Li , we give Maschke＇s theorem for twisted smash products of H-commutative algebras. Through the research on integral and the projective properties of H, charicterized the semisimplicity of twisted smash products A # H. By modular theory of Hopf algebras and the properties of bimodule algebras, for any left-module M and M, defined the structure of right A-module, checked M is A-A bimodule and discussed the properties and module actions of morphism set in a A # H-modules category, we prove that HomA （M, N） is a left A # H-module, thus we obtain H. Furthermore, we study the projective properties of A. Assume that A is semisimple, we obtain that A # H is semisimple if and only if A is projective as a left A # H-module. Finally, under the condition of A is semisimple, we obtain A # H is semisimple if and only if H for some H.

关 键 词：MASCHKE型定理 扭曲冲积 H-交换代数 

分 类 号：O153.3[理学—数学]