Some sufficient and necessary conditions are given for the equivalence between two crossed product actions of Hopf algebra H on the same linear category, and the Maschke theorem is generalized. Based on the result of ...Some sufficient and necessary conditions are given for the equivalence between two crossed product actions of Hopf algebra H on the same linear category, and the Maschke theorem is generalized. Based on the result of the crossed product in the classic Hopf algebra theory, first, let A be a k-linear category and H be a Hopf algebra, and the two crossed products A#_σH and A#'_σH are isomorphic under some conditions. Then, let A#_σH be a crossed product category for a finite dimensional and semisimple Hopf algebra H. If V is a left A#σH-module and WC V is a submodule such that W has a complement as a left A-module, then W has a complement as a A#_σH-module.展开更多
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 ...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.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.11171183)the Shandong Provincial Natural Science Foundation of China(Grant No.ZR2011AM013)
基金The National Natural Science Foundation of China(No.11371088)the Natural Science Foundation of Jiangsu Province(No.BK2012736)+1 种基金the Fundamental Research Funds for the Central Universitiesthe Research Innovation Program for College Graduates of Jiangsu Province(No.KYLX15_0109)
文摘Some sufficient and necessary conditions are given for the equivalence between two crossed product actions of Hopf algebra H on the same linear category, and the Maschke theorem is generalized. Based on the result of the crossed product in the classic Hopf algebra theory, first, let A be a k-linear category and H be a Hopf algebra, and the two crossed products A#_σH and A#'_σH are isomorphic under some conditions. Then, let A#_σH be a crossed product category for a finite dimensional and semisimple Hopf algebra H. If V is a left A#σH-module and WC V is a submodule such that W has a complement as a left A-module, then W has a complement as a A#_σH-module.
基金The National Natural Science Foundation of China(No.11871144,11901240).
文摘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.