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...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.展开更多
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 ...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.展开更多
In Artin algebra representation theory there is an important result which states that when the order of G is invertible in Λ then gl.dim(ΛG)=gl.dim(Λ). With the development of Hopf algebra theory, this result is ge...In Artin algebra representation theory there is an important result which states that when the order of G is invertible in Λ then gl.dim(ΛG)=gl.dim(Λ). With the development of Hopf algebra theory, this result is generalized to smash product algebra. As known, weak Hopf algebra 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.展开更多
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-...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.展开更多
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...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.展开更多
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 rin...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.展开更多
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 ...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.展开更多
Let H be a semisimple Hopf algebra over a field of characteristic 0, and A a finite-dimensional transitive H-module algebra with a 1-dimensional ideal. It is proved that the smash product A#H is isomorphic to a full m...Let H be a semisimple Hopf algebra over a field of characteristic 0, and A a finite-dimensional transitive H-module algebra with a 1-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.展开更多
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 determin...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.展开更多
Let G be an arbitrary group with identity e. Let A be a G-graded ring with identity 1, i. e. A= A<sub>g</sub> （direct sums of additive groups）, with property A<sub>g</sub>A<sub>h</su...Let G be an arbitrary group with identity e. Let A be a G-graded ring with identity 1, i. e. A= A<sub>g</sub> （direct sums of additive groups）, with property A<sub>g</sub>A<sub>h</sub> A<sub>gh</sub>. The smash prod-展开更多
Let G be a group, HG and R a G graded ring. We study the duality Theorem for G actions and smash products R#G/H of the G graded ring R and the G set G/H.
基金This work is supported by National Natural Science Foundation of Chinaby the excellent doctorate fund of Nanjing agricultural university
文摘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.
基金Supported by the Educational Ministry Science Technique Research Key Foundation of China (108154)the National Natural Science Foundation of China (10871170)
文摘This paper gives a duality theorem for weak L-R smash products, which extends the duality theorem for weak smash products given by Nikshych.
基金Supported by the Ningbo Natural Science Foundation(2006A610089)
文摘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.
基金Project supported by the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (No. 704004), the Program for New Century Excellent Talents in Univer-sity (No. 04-0522), and the Natural Science Foundation of Zhejiang Province (No. 102028), China
文摘In Artin algebra representation theory there is an important result which states that when the order of G is invertible in Λ then gl.dim(ΛG)=gl.dim(Λ). With the development of Hopf algebra theory, this result is generalized to smash product algebra. As known, weak Hopf algebra 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.
基金supported by the NSF of China(Nos.11801515,12071441)the Natural Science Foundation of Zhejiang Province(No.LY20A010003)the Foundation of Zhejiang Educational Committee(No.Y201942625).
文摘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.
文摘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.
基金Project (19501007) supported by the National Natural Science Foundation of China
文摘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.
文摘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.
基金supported by the National Natural Science Foundation of China(No.10731070)
文摘Let H be a semisimple Hopf algebra over a field of characteristic 0, and A a finite-dimensional transitive H-module algebra with a 1-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.
文摘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.
文摘Let G be an arbitrary group with identity e. Let A be a G-graded ring with identity 1, i. e. A= A<sub>g</sub> （direct sums of additive groups）, with property A<sub>g</sub>A<sub>h</sub> A<sub>gh</sub>. The smash prod-
文摘Let G be a group, HG and R a G graded ring. We study the duality Theorem for G actions and smash products R#G/H of the G graded ring R and the G set G/H.
