Maschke’s theorem for smash products of quasitriangular weak Hopf algebras

The paper is concerned with the semisimplicity of smash products of quasitriangular weak Hopf algebras. Let (H,R) be a finite dimensional quasitriangular weak Hopf algebra over a field k and A any semisimple and quantum commutative weak H-module algebra. Based on the work of Nikshych et al. (Topol. Appl. 127(1–2):91–123, 2003), we give Maschke’s theorem for smash products of quasitriangular weak Hopf algebras, stating that A#H is semisimple if and only if A is a projective left A#H-module, which extends the Theorem 3.2 given in Yang and Wang (Commun. Algebra 27(3):1165–1170, 1999).     

Structure Theorems for Bicomodule Algebras Over Quasi-Hopf Algebras,Weak Hopf Algebras,and Braided Hopf Algebras

Let H be a quasi-Hopf algebra, a weak Hopf algebra, or a braided Hopf algebra. Let B be an H-bicomodule algebra such that there exists a morphism of H-bicomodule algebras v: H → B. Then we can define an object B^co(H), which is a left-left Yetter–Drinfeld module over H, having extra properties that allow to make a smash product B^co(H)#H, which is an H-bicomodule algebra, isomorphic to B.     

A Morita Context and Galois Extensions for Quasi-Hopf Algebras

If H is a finite dimensional quasi-Hopf algebra and A is a left H-module algebra, we show that there is a Morita context connecting the smash product A#H and the subalgebra of invariants A ^H . We define also Galois extensions and prove the connection with this Morita context, as in the Hopf case.     

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.     

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.     

The Structure Theorem of Weak Comodule Algebras

This article mainly gives the structure theorem of weak comodule algebras, that is, assume that H is a weak Hopf algebra, and B a weak right H-comodule algebra, if there exists a morphism φ: H → B of a weak right H-comodule algebras, then there exists an algebra isomorphism: B ? B ^coH #H, where B ^coH denotes the coinvariant subalgebra of B, and B ^coH #H denotes the weak smash product.     

On Weak Crossed Products of Weak Hopf Algebras

Let H be a weak Hopf algebra in the sense of Böhm et al. (J Algebra 221:385–438, 1999) measuring an algebra A. Let A# _σ H be a weak crossed product with σ invertible. Then in this paper we first give some conditions for A# _σ H to be a weak Hopf algebra. Next the spectral sequence for Ext will be constructed which yields an estimate for the global dimension of A# _σ H in terms of the corresponding data for H and A. Furthermore, we will investigate when A???A# _σ H becomes a separable extension. Finally, we prove that if H and its dual H* are both semisimple, then the finitistic dimension of A# _σ H is equal to that of A.     

On Characteristic Modules of Graded Quasi-hereditary Algebras

Abstract

Let G be a group and A a G-graded quasi-hereditary algebra. Then its characteristic module is proved to be G-gradable, i.e., it is isomorphic to a G-graded module as A-modules. This implies that the Ringel dual A′ of A admits a canonical G-grading which extends to the graded situation the typical equivalence between Δ-good and ?-good modules of A and A′, respectively. It follows some consequences: the derived category of finitely generated G-graded A-modules is equivalent to the derived category of finitely generated G-graded A'-modules; if G is finite, then the Ringel dual of the smash product A#G* is isomorphic to the smash product A'#G* of A' with G.     

The global dimension of L-R twisted smash products

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.     

Morita context of weak Hopf algebras

Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A~H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context connecting the smash product A#H and the invariant subalgebra A~H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.     

A Sufficient Condition for a Smash Product as a Transfinite Left Free Normalizing Extension and Its Application

In this paper, a sufficient condition is given under which the smash product A#H is a transfinite left free normalizing extension of an algebra A. Moreover, the result is applied to a skew semigroup ring, a skew group ring and the quantum group U _q (sl(2)) such that some properties are shown. Received April 14, 1997, Accepted August 10, 1998     

Cleft extensions of Koszul twisted Calabi–Yau algebras

Let H be a twisted Calabi–Yau (CY) Hopf algebra and σ a 2-cocycle on H. Let A be an N-Koszul twisted CY algebra such that A is a graded H^σ- module algebra. We show that the cleft extension A#_σH is also a twisted CY algebra. This result has two consequences. Firstly, the smash product of an N-Koszul twisted CY algebra with a twisted CY Hopf algebra is still a twisted CY algebra. Secondly, the cleft objects of a twisted CY Hopf algebra are all twisted CY algebras. As an application of this property, we determine which cleft objects of U(D, λ), a class of pointed Hopf algebras introduced by Andruskiewitsch and Schneider, are Calabi–Yau algebras.     

On twisted smash products for bimodule algebras and the drinfeld double 1

In this paper we construct a new algebra AHof an H- bimodule algebra Aby a Hopf algebra Hand study some of its properties. The smash product, the Drinfel'd double D(H) and the Doi-Takeuchi's algebra B?,H, are all special cases of AH. Moreover,we find a necessary and sufficient condition for A Hto be a Hopf algebra and also consider the dual situation     

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.     

Derived representation type and Gorenstein projective modules of an algebra under crossed product

Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1-13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.     

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.     

H*－有理模和右Smash积A＃HRH*

本文对Ｈ^*上的有理模Ｍ做了一些讨论，刻划了此类模的某些性质，并利用这些性质得到了右Ｓｍａｓｈ积Ａ＃_H^R［ｋＧ］^*上模Ｍ是完全可约模的条件。     

The complexity of crossed products

Let H be a finite-dimensional Hopf algebra, let A be a finite-dimensional algebra measured by H, and let A # _σ H be a crossed product. In this paper, we first show that if H is semisimple as well as its dual H*, then the complexity of A # _σ H is equal to that of A. Furthermore, we prove that the complexity of a finite-dimensional Hopf algebra H is equal to the complexity of the trivial module _H k. As an application, we prove that the complexity of Sweedler’s 4-dimensional Hopf algebra H ₄ is equal to 1.     

The global dimensions of crossed products and crossed coproducts

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）.     

Semidirect products of weak multiplier Hopf algebras: Smash products and smash coproducts

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.