With the main motivation to present some alternatives to the cosemisimplicity of twisted Smash coproduct,this paper first gives some basic constructions which make sense to consider the derived functors of some left e...With the main motivation to present some alternatives to the cosemisimplicity of twisted Smash coproduct,this paper first gives some basic constructions which make sense to consider the derived functors of some left exact functors from the category of comodules to vector spaces.Then,the Hochschild–Serre spectral sequence for twisted Smash coproduct coalgebras is obtained,which extends some results on homological properties given in the category of comodules over Smash coproduct in the literature.展开更多
设H是一个双代数,B是带有H弱作用的代数,σ:H(?)H→B和τ:H(?)H→B都是k-双线性映射.首先我们给出了B_χ^(#_σ^τ)H成为双代数的充分必要条件,此双代数带有扭曲交叉积B#_σ~τH和冲余积B×H,其中B是H上的余模余代数.此双代数是由Ra...设H是一个双代数,B是带有H弱作用的代数,σ:H(?)H→B和τ:H(?)H→B都是k-双线性映射.首先我们给出了B_χ^(#_σ^τ)H成为双代数的充分必要条件,此双代数带有扭曲交叉积B#_σ~τH和冲余积B×H,其中B是H上的余模余代数.此双代数是由Radford首次在文献[8]中提出,后来Doi and Takeuchi又在文献[4]和[9]中进一步推广而得到的.然后我们对此双代数进行刻画并研究其基本性质.最后我们给出了此双代数成为Hopf代数的充分条件.展开更多
We develop the Radford's biproduct theorem which plays an important role in giving a negative answer to a conjecture of I Kaplansky. Let B, H be two Hopf algebras with H acting weakly on B and α, β : B → H H be...We develop the Radford's biproduct theorem which plays an important role in giving a negative answer to a conjecture of I Kaplansky. Let B, H be two Hopf algebras with H acting weakly on B and α, β : B → H H be two linear maps verifying suitable conditions. We consider in this paper a twisted Hopf crossed coproduct B ×βα H and derive a necessary and sufficient condition for B # ×βα H with a Hopf smash product structure to be a bialgebra which generalizes in [14, Theorem 1.1] and the well-known Radford biproduct theorem [10, Theorem 1] .展开更多
In this paper, we show that if H is a finite-dimensional Hopf algebra such that H and H^＊ are semisimple, then gl.dim（A#σH）=gl.dim（A）, where a is a convolution invertible cocycle. We also discuss the relationshi...In this paper, we show that if H is a finite-dimensional Hopf algebra such that H and H^＊ are semisimple, then gl.dim（A#σH）=gl.dim（A）, where a is a convolution invertible cocycle. We also discuss the relationship of global dimensions between the crossed product A^#σH and the algebra A, where A is coacted by H. Dually, we give a sufficient condition for a finite dimensional coalgebra C and a finite dimensional semisimple Hopf algebra H such that gl.dim（C α H）=gl.dim（C）.展开更多
We present a general construction producing a unitary corepresentation of a multiplier Hopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopf algebra. Then we investigate som...We present a general construction producing a unitary corepresentation of a multiplier Hopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopf algebra. Then we investigate some properties of functors, integrals and morphisms related to the categories of the crossed modules and covariant modules over multiplier Hopf algebras. By doing so, we can apply our theory to the case of group-cograded multiplier Hopf algebras, in particular, the case of Hopf group-coalgebras.展开更多
基金Supported by National Natural Science Foundation of China(10871170)Educational Minister Science Technology Key Foundation of China(10871170)College Special Research Doctoral Disciplines Point Fund of China(20100097110040)
基金Supported by the Natural Science Foundation of Shandong Provience(Grant No.ZR2012AL02)。
文摘With the main motivation to present some alternatives to the cosemisimplicity of twisted Smash coproduct,this paper first gives some basic constructions which make sense to consider the derived functors of some left exact functors from the category of comodules to vector spaces.Then,the Hochschild–Serre spectral sequence for twisted Smash coproduct coalgebras is obtained,which extends some results on homological properties given in the category of comodules over Smash coproduct in the literature.
基金Supported by the FNS of China(10871042)the SRFDPHE(20060286006)the EMSTRKF of China(108154).
文摘设H是一个双代数,B是带有H弱作用的代数,σ:H(?)H→B和τ:H(?)H→B都是k-双线性映射.首先我们给出了B_χ^(#_σ^τ)H成为双代数的充分必要条件,此双代数带有扭曲交叉积B#_σ~τH和冲余积B×H,其中B是H上的余模余代数.此双代数是由Radford首次在文献[8]中提出,后来Doi and Takeuchi又在文献[4]和[9]中进一步推广而得到的.然后我们对此双代数进行刻画并研究其基本性质.最后我们给出了此双代数成为Hopf代数的充分条件.
基金Supported by the NNSF of China(10871042)Supported by the Foster Foundation of Henan Normal University(2010PL01)Supported by the Research Fund of PhD(1005)
文摘We develop the Radford's biproduct theorem which plays an important role in giving a negative answer to a conjecture of I Kaplansky. Let B, H be two Hopf algebras with H acting weakly on B and α, β : B → H H be two linear maps verifying suitable conditions. We consider in this paper a twisted Hopf crossed coproduct B ×βα H and derive a necessary and sufficient condition for B # ×βα H with a Hopf smash product structure to be a bialgebra which generalizes in [14, Theorem 1.1] and the well-known Radford biproduct theorem [10, Theorem 1] .
基金Supported by the National Natural Science Foundation of China (Grant No. 10771182)Nantong University Foundation (Grant No. xj06Z009)
文摘In this paper, we show that if H is a finite-dimensional Hopf algebra such that H and H^＊ are semisimple, then gl.dim（A#σH）=gl.dim（A）, where a is a convolution invertible cocycle. We also discuss the relationship of global dimensions between the crossed product A^#σH and the algebra A, where A is coacted by H. Dually, we give a sufficient condition for a finite dimensional coalgebra C and a finite dimensional semisimple Hopf algebra H such that gl.dim（C α H）=gl.dim（C）.
基金Supported by National Natural Science Foundation of China (Grant No. 10871042) the Research Fund for the Doctoral Program of Higher Education (Grant No. 20060286006)+1 种基金 Natural Science Foundation of Jiangsu Province (Grant No. BK2009258)the Key Project of Chinese Ministry of Education of China (Grant No.108154)
文摘We present a general construction producing a unitary corepresentation of a multiplier Hopf algebra in itself and study RR-corepresentations of twisted tensor coproduct multiplier Hopf algebra. Then we investigate some properties of functors, integrals and morphisms related to the categories of the crossed modules and covariant modules over multiplier Hopf algebras. By doing so, we can apply our theory to the case of group-cograded multiplier Hopf algebras, in particular, the case of Hopf group-coalgebras.
