On the Lie's Theorem for Lie Color Algebras

We show that the Lie's Theorem holds for Lie color algebras with a torsion-free abelian group $G$. We give an example to show that the torsion-free condition is necessary.Comment: 5 page

The Von Neumann Regular Radical and Jacobson Radical of Crossed Products

We construct the $H$-von Neumann regular radical for $H$-module algebras and show that it is an $H$-radical property. We obtain that the Jacobson radical of twisted graded algebra is a graded ideal. For twisted $H$-module algebra $R$, we also show that r_{j}(R#_\sigma H)= r_{Hj}(R)#_\sigma H and the Jacobson radical of $R$ is stable, when $k$ is an algebraically closed field or there exists an algebraic closure $F$ of $k$ such that $r_j(R \otimes F) = r_j(R) \otimes F$, where $H$ is a finite-dimensional, semisimple, cosemisimple, commutative or cocommutative Hopf algebra over $k$. In particular, we answer two questions J.R.Fisher asked.Comment: 15page

Homological Dimension of Crossed Products

We obtain that the global dimensions of $R$ and the crossed product R # _\sigma H are the same; meantime, their weak dimensions are also the same, when $H$ is finite-dimensional semisimple and cosemisimple Hopf algebra.Comment: 13pag

The Radicals of Crossed Products

The relations between the radical of crossed product R #_\sigma H and algebra $R$ are obtained. Using this theory, the author shows that if $H$ is a finite-dimensional semisimple, cosemisimle, and either commutative or cocommutative Hopf algebra, then $R$ is $H$-semiprime iff $R$ is semiprime iff R#_{\sigma}H is semiprime.Comment: 24page

The relation between the decomposition of comodules and coalgebras

T. Shudo and H. Miyamito \cite{SM78} showed that $C$ can be decomposed into a direct sum of its indecomposable subcoalgebras of $C$. Y.H. Xu \cite {XF92} showed that the decomposition was unique. He also showed that $M$ can uniquely be decomposed into a direct sum of the weak-closed indecomposable subcomodules of $M$(we call the decomposition the weak-closed indecomposable decomposition) in \cite{XSF94}. In this paper, we give the relation between the two decomposition. We show that if $M$ is a full, $W$-relational hereditary $C$-comodule, then the following conclusions hold: (1) $M$ is indecomposable iff $C$ is indecomposable; (2) $M$ is relative-irreducible iff $C$ is irreducible; (3) $M$ can be decomposed into a direct sum of its weak-closed relative-irreducible subcomodules iff $C$ can be decomposed into a direct sum of its irreducible subcoalgebras. We also obtain the relation between coradical of $C$- comodule $M$ and radical of algebra $C(M)^*$Comment: 17page

Duality Theorem and Drinfeld Double in Braided Tensor Categories

Let $H$ be a finite Hopf algebra with $C_{H,H} = C_{H,H}^{-1}.$ The duality theorem is shown for $H$, i.e., (R # H)# H^{\hat *} \cong R \otimes (H \bar \otimes H^{\hat *}) \hbox {as algebras in} {\cal C}. Also, it is proved that the Drinfeld double $(D(H),[b])$ is a quasi-triangular Hopf algebra in ${\cal C}$.Comment: 8. to appear in Algebra Colloquiu

The Radicals of Hopf Module Algebras

The characterization of $H$-prime radical is given in many ways. Meantime, the relations between the radical of smash product R # H and the $H$-radical of Hopf module algebra $R$ are obtained.Comment: 18page

Double Bicrosssum of Braided Lie algebras

The condition for double bicrosssum to be a braided Lie bialgebra is given. The result generalizes quantum double, bicrosssum, bicrosscosum, bisum. The quantum double of braided Lie bialgebras is constructed. The relation between double crosssum of Lie algebras and double crossproduct of Hopf algebras is given.Comment: 25pages. Improve some word

On Pointed Hopf Algebras with Sporadic Simple Groups HS{\rm HS} and Co3{\rm Co3}

Every non quasi- -1-type Nichols algebra is infinite dimensional. All quasi- -1-type Nichols algebra over sporadic simple groups ${\rm HS}$ and ${\rm Co3}$ are found.Comment: 9page

The Factorization of Braided Hopf Algebras

We obtain the double factorization of braided bialgebras or braided Hopf algebras, give relation among integrals and semisimplicity of braided Hopf algebra and its factors.Comment: 12page