Twisted smash product for Hopf quasigroups

In order to study algebraic structures of parallelizable sphere s7, the notions of quasimodules and biquasimodnle algebras over Hopf quasigroups, which are not required to be associative, are introduced. The lack of associativity of quasimodules is compensated for by conditions involving the antipode. The twisted smash product for Hopf quasigroups is constructed using biquasimodule algebras, which is a generalization of the twisted smash for Hopf algebras. The twisted smash product and tensor coproduct is turned into a Hopf quasigroup if and only if the following conditions （h1→a） h2 = （h2→a） h1, （a←S（h1）） h2 = （a←S（h2）） h1, hold. The obtained results generalize and improve the corresponding results of the twisted smash product for Hopf algebras.

The smash products of entwining structure

Let k be a commutative ring, C a projective k-coalgebra. The smash productsof entwining structure (A, C)_ψ are discussed. When the map ψ is a bijective, and C is a finitelygenerated k-module, a version of the Ulbrich theorem for coalgebras C is given.

TWISTED PRODUCTS AND SMASH PRODUCTS OVER WEAK HOPF ALGEBRAS

This paper gives a sufficient and necessary condition for twisted products to be weak Hopf algebras, moreover, gives a description for smash products to be weak Hopf algebras. It respectively generalizes R.K.Molnar's major result and I.Boca's result.

模与G-集的Smash Products

对任意可迁 G-集 A,讨论了 G-分次环与 A的 Smash Products,定义并研究了 G-分次环与 G-集的Smash Products,推广了关于分次迹 ,余迹的 Smash

Duality theorem for weak L-R smash products

This paper gives a duality theorem for weak L-R smash products, which extends the duality theorem for weak smash products given by Nikshych.

Global dimension of weak smash product

In Artin algebra representation theory there is an important result which states that when the order of G is invertible in A then gl.dim（AG）=gl.dim（A）. With the development of Hopf algebra theory, this result is generalized to smash product algebra. As known, weak Hopfalgebra is an important generalization of Hopf algebra. In this paper we give the more general result, that is the relation of homological dimension between an algebra A and weak smash product algebra A#H, where H is a finite dimensional weak Hopf algebra over a field k and A is an H-module algebra.

The Relative Properties of Graded Ring R and Smash Product R # G

Led G he a group, R be a G-graded ring, R,this paper deals with some relative properties of R and smash product R # G .We give out R is graded regular ring,graded right SF-ring, graded right V-ring, graded QF-ring if and only if R # G is regular ring,right SF-rign, right V-ring and QF-ring respectively.

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 Product的有限性条件

本文借助群分次环理论,部分刻划Smash product A#G的有限性条件(包括半单的、Artin的、Noether的,遗传的等)。

On BiHom-L-R Smash Products

Let(H,αH,βH,ψH,ωH,SH)be a BiHom-Hopf algebra and(A,αA,βA)be an(H,αH,βH,ψH,ωH)-BiHom-bimodule algebra,where the mapsαH,βH,ψH,ωH,αA,βA are bijective.We first prove the Maschke-type theorem for the BiHom-L-R smash product over a finite-dimensional semisimple BiHom-Hopf algebra.Next we give a Morita context between the BiHom-subalgebra A^(biH)and the BiHom-L-R smash product A#H.

Weak Global Dimension of Smash Products of Hopf Algebras

Let H be a finite dimensional Hopf algebra and A a commutative H-module algebra. We prove that the smash product A#H is of the same weak global homological dimension as A, provided that H^＊ is unimodular and there is a trace one element in A.

模的Smash Product及分次迹和分次余迹

在群分次环的分次模范畴上定义了分次迹、分次余迹和分次模的ＳｍａｓｈＰｒｏｄｕｃｔ，证明了关于迹与余迹的许多结论对分次迹与分次余迹仍成立．对任一分次模Ｍ，给出了Ｍ的分次迹、分次余迹的ＳｍａｓｈＰｒｏｄｕｃｔ与Ｍ的ＳｍａｓｈＰｒｏｄｕｃｔ的迹和余迹之间的关系，从而统一和推广了关于分次Ｂｅａｒ根及分次Ｊａｃｏｂｓｏｎ根的ＳｍａｓｈＰｒｏｄｕｃｔ的刻划，同时还刻划了ＳｍａｓｈＰｒｏｄｕｃｔ函子与函子（）＊之间的关系．

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.

THE GENERALIZED SMASH PRODUCT AND COPRODUCT

The author discusses the braiding structures of the generalized smash product bialgebra and the cobraiding structures of the generalized smash coproduct bialgebra. It is pointed out that doublecrossed product determined by a cocycle is the generalized smash product and that doublecocrossed coproduct determined by a weak R-matrix is the generalized smash coproduct.

L-R-Smash Products and L-R-Twisted Tensor Products of Algebras

We introduce a common generalization of the L-R-smash product and twisted tensor product of algebras, under the name L-R-twisted tensor product of algebras. We investigate some properties of this new construction, for instance, we prove a result of the type ＂invariance under twisting＂ and we show that under certain circumstances L-R- twisted tensor products of algebras may be iterated.

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.

Generalized Smash Products

In this paper, we study the ring #(D,B) and obtain two very interesting results. First we prove in Theorem 3 that the category of rational left BU-modules is equivalent to both the category of #-rational left modules and the category of all (B,D)-Hopf modules D . Cai and Chen have proved this result in the case B = D = A. Secondly they have proved that if A has a nonzero left integral then A#A *rat is a dense subring of End k (A). We prove that #(A,A) is a dense subring of End k (Q), where Q is a certain subspace of #(A,A) under the condition that the antipode is bijective (see Theorem 18). This condition is weaker than the condition that A has a nonzero integral. It is well known the antipode is bijective in case A has a nonzero integral. Furthermore if A has nonzero left integral, Q can be chosen to be A (see Corollary 19) and #(A,A) is both left and right primitive. Thus A#A *rat ? #(A,A) ? End k (A). Moreover we prove that the left singular ideal of the ring #(A,A) is zero. A corollary of this is a criterion for A with nonzero left integral to be finite-dimensional, namely the ring #(A,A) has a finite uniform dimension.

On the Semiprimitivity and Semiprimality Problems for Partial Smash Products

In this paper we discuss about the semiprimitivity and the semiprimality of partial smash products. Let H be a semisimple Hopf algebra over a field k and let A be a left partial H-module algebra. We study the H-prime and the H-Jacobson radicals of A and their relations with the prime and the Jacobson radicals of A#H, respectively. In particular, we prove that if A is H-semiprimitive, then A^fH is semiprimitive provided that all irreducible representations of A are finite-dimensional, or A is an affine PI-algebra over k and k is a perfect field, or A is locally finite. Moreover, we prove that A=#=H is semiprime provided that A is an H-semiprime PI-algebra, generalizing to the setting of partial actions the known results for global actions of Hopf algebras.

Maschke-Type Theorem for Two-Sided Weak Smash Products

This paper mainly gives a Maschke-type theorem for two-sided weak smash products over semisimple weak Hopf algebras.