Crossed products for Hopf group-algebras  

Hopf群代数上的交叉积

作　　者：You Miman Lu Daowei Wang Shuanhong 游弥漫;鹿道伟;王栓宏(华北水利水电大学数学与统计学院,郑州450045;济宁学院数学系,曲阜273155;东南大学数学学院,南京211189)

机构地区：[1]School of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450045, China [2]Department of Mathematics, Jining University, Qufu 273155, China [3]School of Mathematics, Southeast University, Nanjing 211189, China

出　　处：《Journal of Southeast University(English Edition)》2021年第3期339-342,共4页东南大学学报（英文版）

基　　金：The National Natural Science Foundation of China(No.11871144,11901240).

摘　　要：First,the group crossed product over the Hopf group-algebras is defined,and the necessary and sufficient conditions for the group crossed product to be a group algebra are given.The cleft extension theory of the Hopf group algebra is introduced,and it is proved that the crossed product of the Hopf group algebra is equivalent to the cleft extension.The necessary and sufficient conditions for the crossed product equivalence of two Hopf groups are then given.Finally,combined with the equivalence theory of the Hopf group crossed product and cleft extension,the group crossed product constructed by the general 2-cocycle as algebra is determined to be isomorphic to the group crossed product of the 2-cocycle with a convolutional invertible map of the 2-cocycle.The unit property of a general 2-cocycle is equivalent to the convolutional invertible map of the 2-cocycle,and the combination condition of the weak action is equivalent to the convolutional invertible map of the 2-cocycle and the combination condition of the weak action.Similarly,crossed product algebra constructed by the general 2-cocycle is isomorphic to the Hopfπ-crossed product algebra constructed by the 2-cocycle with a convolutional invertible map.首先给出了Hopf群代数的群交叉积定义,并给出了群交叉积是群代数的充分必要条件.引入了Hopf群代数的cleft扩张理论,并证明了Hopf群代数的交叉积与cleft扩张等价.然后,给出了2个Hopf群交叉积等价的充分必要条件.最后,结合Hopf群交叉积与cleft扩张的等价理论得到,群文叉积一般由2-余循环构造,作为代数同构于带有卷积可逆映射的2-余循环的群交叉积.一般2-余循环的余单位性质等价于带有卷积可逆映射的2-余循环余单位性质,通常意义下的2-余循环和弱作用结合条件等价于带有卷积可逆映射的2-余循环及其弱作用结合条件;同时得到,由一般2-余循环构造的Hopfπ-交叉积代数同构于带有卷积可逆映射的2-余循环构造的Hopfπ-交叉积代数.

关 键 词：Hopfπ-algebra cleft extension theorem π-comodule-like algebra group crossed products 

分 类 号：O15[理学—数学]