Construction of semisimple category over generalized Yetter-Drinfeld modules

Let H be a commutative, noetherian, semisimple and cosemisimple Hopf algebra with a bijective antipode over a field k. Then the semisimplicity of YD（H） is considered, where YD （H） means the disjoint union of the category of generalized Yetter-Drinfeld modules nYD^H（ α, β） for any α, β E Aut Hopf（H）. First, the fact that YD（H） is closed under Mor is proved. Secondly, based on the properties of finitely generated projective modules and semisimplicity of H, YD（H） satisfies the exact condition. Thus each object in YD（H） can be decomposed into simple ones since H is noetherian and cosemisimple. Finally, it is proved that YD （H） is a sernisimple category.

The Weyl’s Theorem and Its Perturbations in Semisimple Banach Algebra

Denote a semisimple Banach algebra with an identity e by A.This paper studies the Fredholm,Weyl and Browder spectral theories in a semisimple Banach algebra,and meanwhile considers the properties of the Fredholm element,the Weyl element and the Browder element.Further,for a∈A,we give the Weyl’s theorem and the Browder’s theorem for a,and characterize necessary and sufficient conditions that both a and f(a)satisfy the Weyl’s theorem or the Browder’s theorem,where f is a complex-valued function analytic on a neighborhood ofσ(a).In addition,the perturbations of the Weyl’s theorem and the Browder’s theorem are investigated.

On Ellipsoids Attached to Root Systems

For any finite-dimensional complex semisimple Lie algebra, two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations, and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group on the sets of Diophantine solutions of the equations of the ellipsoids. The primary realization of the Weyl group suggests an order on the Weyl group, which is stronger than the Chevalley-Bruhat ordering of the Weyl group, and which provides an algorithm for the Chevalley-Bruhat ordering. The secondary realization of the Weyl group provides an algorithm for constructing all reduced expressions for any of its elements, and thus provides another way for the Chevalley-Bruhat ordering of the Weyl group.

On Smash Products of Transitive Module Algebras

Let H be a semisimple Hopf algebra over a field of characteristic 0, and A a finite-dimensional transitive H-module algebra with a l-dimensional ideal. It is proved that the smash product A#H is isomorphic to a full matrix algebra over some right coideal subalgebra N of H. The correspondence between A and such N, and the special case A = k（X） of function algebra on a finite set X are considered.

On Hopf Galois Extension of Separable Algebras

In this paper, the classical Galois theory to the H＊-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra, and A/An a right H＊-Galois extension. The authors prove that, if An is a separable kalgebra, then for any right coideal subalgebra B of H, the B-invariants AB = {a ∈ A ｜ b · a = ε（b）a, Ab ε B} is a separable k-algebra. They also establish a Galois connection between right coideal subalgebras of H and separable subalgebras of A containing AH as in the classical case. The results are applied to the case H = （kG）＊ for a finite group G to get a Galois 1-1 correspondence.

The centralizer of an element in a Lie algebra of type L

In this paper, we investigate the Lie algebra L(A,α,δ) of type L and obtain the respective sufficient conditions for L(A,α,δ δ to be semisimple, and for Z(ω) = Fω as well, where 0 ≠ ω ? L(A, α, δ, δ) and Z(ω) is the centralizer of ω.

The Semisimplicity of L^i(K,w) of a Weighted Commutative Hypergroup K

In the present paper, it is shown that, for a locally compact commutative hypergroup K with a Borel measurable weight function w, the Banach algebra L^1 （K, w） is semisimple if and only if L^1 （K） is semisimple. Indeed, we have improved compact groups to the general setting of locally a well-krown result of Bhatt and Dedania from locally compact hypergroups.