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1.
In this paper we generalize the notions of crossed products and L-R smash products in the context of multiplier Hopf algebras. We use comodule algebras to define generalized diagonal crossed products, L-R smash products, two-sided smash products and two-sided crossed products and prove that they are all associative algebras. Then we show the isomorphic relations of them.  相似文献   

2.
Wei Wang  Nan Zhou 《代数通讯》2018,46(8):3241-3261
In this paper, we will develop the smash product of weak multiplier Hopf algebras unifying the cases of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras. We will show that the smash product R#A has a regular weak multiplier Hopf algebra structure if R and A are regular weak multiplier Hopf algebras. We shall investigate integrals on R#A. We also consider the result in the ?-situation and new examples. Dually, we consider the smash coproduct of weak multiplier Hopf algebras under an appropriate form and integrals on the smash coproduct and we obtain results in the ?-situation.  相似文献   

3.
In this paper, the notion of L-R crossed coproduct is introduced as a unified approach for smash coproducts, crossed coproducts and L-R smash coproducts of Hopf algebras. A duality theorem for L-R crossed coproduct is proved.  相似文献   

4.
In this paper, we introduce a generalized Hopf Galois theory for regular multiplier Hopf algebras with integrals, which might be viewed as a generalization of the Hopf Galois theory of finite-dimensional Hopf algebras. We introduce the notion of a coaction of a multiplier Hopf algebra on an algebra. We show that there is a duality for actions and coactions of multiplier Hopf algebras with integrals. In order to study the Galois (co)action of a multiplier Hopf algebra with an integral, we construct a Morita context connecting the smash product and the coinvariants. A Galois (co)action can be characterized by certain surjectivity of a canonical map in the Morita context. Finally, we apply the Morita theory to obtain the duality theorems for actions and coactions of a co-Frobenius Hopf algebra.  相似文献   

5.
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.  相似文献   

6.
《代数通讯》2013,41(9):3403-3415
Abstract

This paper, mainly gives a Fundamental Theorem of weak Doi–Hopf modules, which is not only generalizes the Fundamental Theorem of weak Hopf modules but also generalizes the Fundamental Theorem of relative Hopf modules. Moreover, it gives a sufficient and necessary condition for weak smash product algebras to be weak bialgebras, and a sufficient condition for weak smash product algebras to be semisimple weak Hopf algebras.  相似文献   

7.
8.
王勇 《数学杂志》2015,35(5):1127-1138
本文研究了Hopf代数胚上的Smash积代数.利用Hopf代数胚的积分理论,获得了Hopf代数胚上的Smash的Maschke型定理并构造了一个Morita关系,推广了Cohen和Fishman在文献[1]中的相应结果.作为应用,获得了Hopf代数胚上的余模代数的Maschke型定理.  相似文献   

9.
在弱Hopf代数上,定义了交叉积概念,并且得到了它的两种特殊形式,冲积和扭积.特别地,给出了扭积为弱Hopf代数的一个充要条件,推广了Hopf代数的相应结论.  相似文献   

10.
弱Hopf代数作用与冲积   总被引:1,自引:0,他引:1  
王志玺  侯波 《数学学报》2007,50(1):89-96
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.  相似文献   

11.
12.
E. J. Beggs  E. J. Taft 《代数通讯》2013,41(10):3511-3523
The Doi–Koppinen generalized smash product can be applied to linear maps on bialgebras, and we define a new associative product for linear maps on bialgebras. We comment on the applications of these products to left Hopf algebras and Hopf algebroids.  相似文献   

13.
We define and study the property finite presentability in the category  of Hopf algebras that are smash product of universal enveloping algebra of a Lie algebra by a group algebra. We show that for such Hopf algebras finite presentability is equivalent with finite presentability as an associative k-algebra.  相似文献   

14.
Xuan Zhou 《代数通讯》2013,41(12):4613-4632
In this article, we mainly study a new notion of a generalized smash product for weak Hopf comodule algebras and provide a new version of the duality theorem for weak smash products as an application.  相似文献   

15.
Pairing and Quantum Double of Multiplier Hopf Algebras   总被引:2,自引:0,他引:2  
We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.  相似文献   

16.
We construct the new algebra A#H of an H-bimodule algebra A called the L-R twisted smash product, and give the duality theorem for L-R twisted smash products which extends the duality theorem for smash products given by Blattner and Montgomery. Furthermore, by using the duality theorem for L-R twisted smash products, we establish the relationship of global dimension between the H-bimodule algebra A and its L-R twisted smash product A#H.  相似文献   

17.
L. Delvaux 《代数通讯》2013,41(1):346-360
In this article we lay the algebraic foundations to establish the existence of trace functions on infinite-dimensional (multiplier) Hopf algebras. We solve the problem within the framework of multiplier Hopf algebra with integrals. By applying this theory to group-cograded multiplier Hopf algebras, we prove the existence of group-traces on group-cograded multiplier Hopf algebras with possibly infinite-dimensional components. We generalize the results as obtained by Virelizier in the case of finite-type Hopf group-coalgebras.  相似文献   

18.
Lihui Zhao  Diming Lu 《代数通讯》2013,41(1):248-272
The goal of this article is to generalize the theory of Hopf–Ore extensions on Hopf algebras to multiplier Hopf algebras. First the concept of a Hopf–Ore extension of a multiplier Hopf algebra is introduced. We give a necessary and sufficient condition for Ore extensions to become a multiplier Hopf algebra. Finally, *-structures are constructed on Hopf–Ore extensions, and certain isomorphisms between Hopf–Ore extensions are discussed.  相似文献   

19.
Let B be a braided Hopf algebra (with bijective antipode) in the category of left Yetter-Drinfeld modules over a quasi-Hopf algebra H. As in the case of Hopf algebras (J. Algebra 92 (1985) 322), the smash product B#H defined in (Comm. Algebra 28(2) (2000) 631) and a kind of smash coproduct afford a quasi-Hopf algebra structure on BH. Using this, we obtain the structure of quasi-Hopf algebras with a projection. Further we will use this biproduct to describe the Majid bosonization (J. Algebra 163 (1994) 165) for quasi-Hopf algebras.  相似文献   

20.
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.  相似文献   

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