Twisted Smash Products and L-R Smash Products for Biquasimodule Hopf Quasigroups

We consider two new algebras from an H-biquasimodule algebra A and a Hopf quasigroup H: twisted smash product A ? H and L-R smash product A?H, and find necessary and sufficient conditions for making them Hopf quasigroups. We generalize the main results in Brzeziński and Jiao [5] and Klim and Majid [9]. Moreover, if H is a cocommutative Hopf quasigroup, we prove that A ? H is isomorphic to A?H as Hopf quasigroups.     

On Smash Products Of Hopf Algebras

Let H = X? _R A denote an R-smash product of the two bialgebras X and A. We prove that (X,A) is a pair of matched bialgebras, if the R-smash product H has a braiding structure. When X is an associative algebra and A is a Hopf algebra, we investigate the global dimension and the weak dimension of the smash product H = X? _R A and show that lD(H) ≤ rD(A) + lD(X) and wD(H) ≤ wD(A) + wD(X). As an application, we get lD(H ₄) = ∞ for Sweedler's four dimensional Hopf algebra H ₄. We also study the associativity of smash products and the relations between smash products and factorization for algebras.     

Hopf代数的冲积的弱整体维数

设H是有限维Hopf代数,A是交换的H-模代数。当H~*是幺模且A中存在迹为1的元素时,本文证明冲积A#H与代数A的弱整体维数相等。     

L-R smash products for multiplier Hopf algebras

The theory of L-R smash product is extended to multiplier Hopf algebras and a sufficient condition for L-R smash product to be regular multiplier Hopf algebras is given. In particular, the result of the paper implies Delvaux＇s main theorem in the case of smash products.     

Smash积,辫积和L-R Smash积的新对偶

本文主要地证明:由H-重模代数A,B构成的Smash积A#B的新对偶H(A#B)~0恰好是由重模余代数_HA~0,_HB~0构成的Smash余积_HA~0×_HB~0;如果(H,σ)是辫化Hopf代数,则新对偶_HH~0是右,左H~0-重模余代数;由量子Yang-Baxter H-模代数A,B构成的辫积AαB的新对偶(AαB)~0恰好是由量子Yang-Baxiter H-模余代数_HA~0,_HB~0构成的辫余积_HA~0×_HB~0.最后它给出由H-双模代数A构成的L-R Smash积A■H的新对偶(A■H)_H~0的正合序列。     

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.     

Hopf代数交叉积的同调维数

Let H be a finite dimensional cocommutative Hopf algebra and A an H-module algebra, ln this paper, we characterize the projectivity (injectivity) of M as a left A#σ H-module when it is projective (injective) as a left A-module. The sufficient and necessary condition for A#σ H, the crossed product, to have finite global homological dimension is given, in terms of the global homological dimension of A and the surjectivity of trace maps, provided that H is cocommutative and A is commutative.     

On Stable Equivalence of Tensor Products of Algebras

The paper investigates the following problem. Let bimodules N, M yield a stable equivalence of Morita type between self-injective K-algebras A and E. Further, let bimodules S, T yield a stable equivalence of Morita type between self-injective K-algebras B and F. Then we want to know whether the functor M ? _A  ? ? _B S: mod(A ? _K B ^op ) → mod(E ? _K F ^op ) induces a stable equivalence between A ? _K B ^op and E ? _K F ^op . There is given a reduction of this problem to some smaller subcategories for self-injective algebras. Moreover, new invariants of stable equivalences of Morita type are constructed in a general case of arbitrary finite-dimensional algebras over a field.     

Gorenstein代数上的Hopf代数作用

令H是有限维Hopf代数，A是左H-模代数。本文证明了A是Gorenstein代数的充分必要条件。A^H也是Gorenstein代数的条件。它是Enochs EE,GarciaJJ和del RioA关于群作用相应的理论的推广，同时给出A/A^H是Frobenius扩张的条件。     

Hopf代数的双交叉积

本文定义并详细讨论了交叉余积，考虑交叉积与交叉余积合起未成为双代数的问题，讨论了由内作用，内余作用构造的双交叉积．我们还用正合序列刻划了双交叉积．     

弱Hopf代数作用与冲积

本文研究了弱Hopf代数上的冲积并讨论了它约性质．设H是弱Hopf代数,A是左H-摸代数．我们给出了冲积A#H是弱双代数的一个充分条件以及A#H是A可分扩张的一个判定条件．另外,利用积分理论研究了Hopf模代数的有限性条件．     

半单弱Hopf代数的作用

令H是半单弱Hopf代数, A是左H-模代数.我们证明了正则A-模的内射维数, A#H-模A的内射维数和正则A#H-模的内射维数三者是相等的. 而且,利用H在A上的不动点代数我们给出了A是Gorenstein代数的充要条件.     

The Factorization Problem and the Smash Biproduct of Algebras and Coalgebras

We consider the factorization problem for bialgebras. Let L and H be algebras and coalgebras (but not necessarily bialgebras) and consider two maps R : H L L H and W : L H H L. We introduce a product K = L _{W R} H and we give necessary and sufficient conditions for K to be a bialgebra. Our construction generalizes products introduced by Majid and Radford. Also, some of the pointed Hopf algebras that were recently constructed by Beattie, Dsclescu and Grünenfelder appear as special cases.     

On Quasi-Bicrossed Products of Weak Hopf Algebras

The aim of this paper is to study quasi-bicrossed products and a especially quantum quasi-doubles. Firstly, we construct one new kind of quasi-bicrossed products by weak Hopf algebras and then devote a brief discussion to this matter. And, we discuss the conditions for quasi-bicrossed products to possess the structure of almost weak Hopf algebras, containing the case of a special smash product. At the end, we give some properties on the quantum quasi-double, respectively on the quasi-R-isomorphism, the representation-theoretic interpretation and the regularity of the quasi-R-matrix.     

Smash Products of H-Simple Module Algebras

Let H be a finite-dimensional Hopf algebra and A a finite-dimensional H-simple left H-module algebra. We show that the smash product A#H is isomorphic to End _A′(V ? H*), where V ≠ 0 is a finite-dimensional left A-module and (A′, V′) the stabilizer of (A, V). As an application it is proved that A#H is isomorphic to a full matrix algebra over A′ when H is semisimple and dim V|dim A.     

Some Aspects of Strongly P-projective Modules and Homological Dimensions

A left R-module M is called strongly P-projective if Extⁱ(M, P) = 0 for all projective left R-modules P and all i ≥ 1. In this article, we first discuss properties of strongly P-projective modules. Then we introduce and study the strongly P-projective dimensions of modules and rings. The relations between the strongly P-projective dimension and other homological dimensions are also investigated.     

弱HOPF代数的中心构建与基本定理

范畴D(H)^M与范畴H^M的中心是同构的，并将Hopf模的结构定理推广到弱Hopf代数上。     

On the Morita Equivalence of Tensor Algebras

We develop a notion of Morita equivalence for general C^*-correspondencesover C^*-algebras. We show that if two correspondences are Moritaequivalent, then the tensor algebras built from them are stronglyMorita equivalent in the sense developed by Blecher, Muhly andPaulsen. Also, the Toeplitz algebras are strongly Morita equivalentin the sense of Rieffel, as are the Cuntz–Pimsner algebras.Conversely, if the tensor algebras are strongly Morita equivalent,and if the correspondences are aperiodic in a fashion that generalizesthe notion of aperiodicity for automorphisms of C^*-algebras,then the correspondences are Morita equivalent. This generalizesa venerated theorem of Arveson on algebraic conjugacy invariantsfor ergodic, measure-preserving transformations. The notionof aperiodicity, which also generalizes the concept of fullConnes spectrum for automorphisms, is explored; its role inthe ideal theory of tensor algebras and in the theory of theirautomorphisms is investigated. 1991 Mathematics Subject Classification:46H10, 46H20, 46H99, 46M99, 47D15, 47D25.     

On Krull and Valuative Dimension of Tensor Products of Algebras

This article is concerned with the dimension theory of tensor products of algebras over a field k. In fact, we provide formulas for the Krull and valuative dimension of A? _k B when A and B are k-algebras such that the polynomial ring A[n] is an AF-domain for some positive integer n. Also, we compute dim _v (A? _k B) in the case where A ? B.     

On Semisimple Hopf Algebras of Dimension 2 m

In this paper we classify all nontrivial semisimple Hopf algebras of dimension 2 ^{n +1} with the group of grouplikes isomorphic to _{2 n–1}×₂. Moreover, we extend some results on irreducible representations from groups to semisimple Hopf algebras and prove that certain semisimple Hopf algebras, including the ones classified in this paper, satisfy the generalized power map property.