Let H be a Hopf algebra and B an algebra with two linear maps δ, τ： H H→B. The necessary and sufficient conditions for the twisted crossed product B#^τδH equipped with the tensor product coalgebra structure to b...Let H be a Hopf algebra and B an algebra with two linear maps δ, τ： H H→B. The necessary and sufficient conditions for the twisted crossed product B#^τδH equipped with the tensor product coalgebra structure to be a bialgebra are proved. Then, B#^τδH is a coquasitriangular Hopf algebra under certain conditions. This coquasitriangular Hopf algerbra generalizes some known cross products. Finally, as an application, an explicit example is given.展开更多
We first prove that for a finite dimensional non-semisimple Hopfalgebra H, the trivial H-module is not projective and so the almost split sequence ended with k exists. By this exact sequence, for all indecomposable H-...We first prove that for a finite dimensional non-semisimple Hopfalgebra H, the trivial H-module is not projective and so the almost split sequence ended with k exists. By this exact sequence, for all indecomposable H-module X, we can construct a special kind of exact sequence ending with it. The main aim of this paper is to determine when this special exact sequence is an almost split one. For this aim, we restrict H to be tmimodular and the square of its antipode to be an inner automorphism. As a special case, we give an application to the quantum double D（H）=（H^op）^＊∞ H） of any non-semisimple Hopf algebra.展开更多
The Leibniz-Hopf algebra is the free associative Z - algebra with one generator in each positive degree and coproduct is given by the Cartan formula. It has been also known as the ＇ring ofnoncommutative symmetric fun...The Leibniz-Hopf algebra is the free associative Z - algebra with one generator in each positive degree and coproduct is given by the Cartan formula. It has been also known as the ＇ring ofnoncommutative symmetric functions＇ [1], and to be isomorphic to the Solomon Descent algebra [ 12]. This Hopf algebra has links with algebra,topology and combinatorics. In this article we consider another approach of proof for the antipode formula in the Leibniz-Hopf algebra by using some properties of words in [2].展开更多
Let H be an arbitrary Hopf algebra over a field k. In this paper, at first wedeal with the relationship between solutions to the Yang-Baxter equation and quantumYang-Baxter H-comodules; then we use the results to give...Let H be an arbitrary Hopf algebra over a field k. In this paper, at first wedeal with the relationship between solutions to the Yang-Baxter equation and quantumYang-Baxter H-comodules; then we use the results to give a solution to the Yang-Baxterequation over H.展开更多
We focus on the classification of pointed p^3-dimensional Hopf algebras H over any algebraically closed field of prime characteristic p 〉 0. In particular, we consider certain cases when the group of grouplike elemen...We focus on the classification of pointed p^3-dimensional Hopf algebras H over any algebraically closed field of prime characteristic p 〉 0. In particular, we consider certain cases when the group of grouplike elements is of order p or p^2 that is, when H is pointed but is not connected nor a group algebra. The structures of the associated graded algebra gr H are completely described as bosonizations of graded braided Hopf algebras over group algebras, and most of the lifting structures of H are given. This work provides many new examples of （parametrized） non-commutative, non-cocommutative finite- dimensional Hopf algebras in positive characteristic.展开更多
基金supported by the NSFC(11201231)the China Postdoctoral Science Foundation(2012M511643)+1 种基金the Jiangsu Planned Projects for Postdoctoral Research Funds(1102041C)Agricultural Machinery Bureau Foundation of Jiangsu Province(GXZ11003)
文摘Let H be a Hopf algebra and B an algebra with two linear maps δ, τ： H H→B. The necessary and sufficient conditions for the twisted crossed product B#^τδH equipped with the tensor product coalgebra structure to be a bialgebra are proved. Then, B#^τδH is a coquasitriangular Hopf algebra under certain conditions. This coquasitriangular Hopf algerbra generalizes some known cross products. Finally, as an application, an explicit example is given.
基金Project (No. 10371107) supported by the National Natural ScienceFoundation of China
文摘We first prove that for a finite dimensional non-semisimple Hopfalgebra H, the trivial H-module is not projective and so the almost split sequence ended with k exists. By this exact sequence, for all indecomposable H-module X, we can construct a special kind of exact sequence ending with it. The main aim of this paper is to determine when this special exact sequence is an almost split one. For this aim, we restrict H to be tmimodular and the square of its antipode to be an inner automorphism. As a special case, we give an application to the quantum double D（H）=（H^op）^＊∞ H） of any non-semisimple Hopf algebra.
文摘The Leibniz-Hopf algebra is the free associative Z - algebra with one generator in each positive degree and coproduct is given by the Cartan formula. It has been also known as the ＇ring ofnoncommutative symmetric functions＇ [1], and to be isomorphic to the Solomon Descent algebra [ 12]. This Hopf algebra has links with algebra,topology and combinatorics. In this article we consider another approach of proof for the antipode formula in the Leibniz-Hopf algebra by using some properties of words in [2].
文摘Let H be an arbitrary Hopf algebra over a field k. In this paper, at first wedeal with the relationship between solutions to the Yang-Baxter equation and quantumYang-Baxter H-comodules; then we use the results to give a solution to the Yang-Baxterequation over H.
文摘We focus on the classification of pointed p^3-dimensional Hopf algebras H over any algebraically closed field of prime characteristic p 〉 0. In particular, we consider certain cases when the group of grouplike elements is of order p or p^2 that is, when H is pointed but is not connected nor a group algebra. The structures of the associated graded algebra gr H are completely described as bosonizations of graded braided Hopf algebras over group algebras, and most of the lifting structures of H are given. This work provides many new examples of （parametrized） non-commutative, non-cocommutative finite- dimensional Hopf algebras in positive characteristic.
