This paper gives a sufficient and necessary condition for a bialgebra to be a Long bialgebra, and proves a braided product to be a Long bialgebra under some conditions. It also gives a direct sum decomposition of quan...This paper gives a sufficient and necessary condition for a bialgebra to be a Long bialgebra, and proves a braided product to be a Long bialgebra under some conditions. It also gives a direct sum decomposition of quantum Yang-Baxter modules over Long bialgebras.展开更多
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
We first prove that,for any generalized Hamiltonian type Lie algebra H,the first co- homology group H^1(H,H(?)H) is trivial.We then show that all Lie bialgebra structures on H are triangular.
Gel'fand-Dorfman bialgebra,which is both a Lie algebra and a Novikov algebra with some compatibility condition,appeared in the study of Hamiltonian pairs in completely integrable systems.They also emerged in the d...Gel'fand-Dorfman bialgebra,which is both a Lie algebra and a Novikov algebra with some compatibility condition,appeared in the study of Hamiltonian pairs in completely integrable systems.They also emerged in the description of a class special Lie conformal algebras called quadratic Lie conformal algebras.In this paper,we investigate the extending structures problem for Gel'fand-Dorfman bialgebras,which is equivalent to some extending structures problem of quadratic Lie conformal algebras.Explicitly,given a Gel'fand-Dorfman bialgebra(A,o,[.,.]),this problem is how to describe and classify all Gel'fand-Dorfman bialgebra structures on a vector space E(A⊂E)such that(A,o,[.,.])is a subalgebra of E up to an isomorphism whose restriction on A is the identity map.Motivated by the theories of extending structures for Lie algebras and Novikov algebras,we construct an object gH2(V,A)to answer the extending structures problem by introducing the notion of a unified product for Gel'fand-Dorfman bialgebras,where V is a complement of A in E.In particular,we investigate the special case when dim(V)=1 in detail.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
Jordan D-bialgebras were introduced by Zhelyabin.In this paper,we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra.Motivated ...Jordan D-bialgebras were introduced by Zhelyabin.In this paper,we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra.Motivated by the essential connection between Lie bialgebras and Manin triples,we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan-Manin triples called double constructions of pseudo-euclidean Jordan algebras.We also show that a Jordan D-bialgebra leads to the Jordan Yang-Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang-Baxter equation corresponds to an antisymmetric bilinear form,which we call a Jordan symplectic form on Jordan algebras.Furthermore,there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form.展开更多
Let M be an exact symplectic manifold with contact type boundary such that cl（M） = O. Motivated by noncommutative symplectic geometry and string topology, we show that the cyclic cohomology of the Fukaya category of...Let M be an exact symplectic manifold with contact type boundary such that cl（M） = O. Motivated by noncommutative symplectic geometry and string topology, we show that the cyclic cohomology of the Fukaya category of M has an involutive Lie bialgebra structure.展开更多
In the present paper, we investigate the dual Lie coalgebras of the centerless W(2,2) algebra by studying the maximal good subspaces. Based on this, we construct the dual Lie bialgebra structures of the centerless W(2...In the present paper, we investigate the dual Lie coalgebras of the centerless W(2,2) algebra by studying the maximal good subspaces. Based on this, we construct the dual Lie bialgebra structures of the centerless W(2,2) Lie bialgebra. As by-products, four new infinite dimensional Lie algebras are obtained.展开更多
It is shown that the dual bialgebra of any quasitriangular bialgebra is braided, and the dual bialgebra of some braided bialgebra is quasitriangular.Also it is proved that every nondegenerate finite dimensional braid...It is shown that the dual bialgebra of any quasitriangular bialgebra is braided, and the dual bialgebra of some braided bialgebra is quasitriangular.Also it is proved that every nondegenerate finite dimensional braided (dually, quasitriangular) bialgebra has an antipode.展开更多
The present paper is a continuation of [1], where we considered braided infinitesimal Hopf algebras (i.e., infinitesimal Hopf algebras in the Yetter-Drin feld category for any Hopf algebra H), and constructed their Dr...The present paper is a continuation of [1], where we considered braided infinitesimal Hopf algebras (i.e., infinitesimal Hopf algebras in the Yetter-Drin feld category for any Hopf algebra H), and constructed their Drinfeld double as a generalization of Aguiar’s result. In this paper we mainly investigate the necessary and sufficient condition for a braided infinitesimal bialgebra to be a braided Lie bialgebra (i.e., a Lie bialgebra in the category ).展开更多
In this paper,the structures of dual Lie bialgebras of twisted SchrSdinger-Virasoro type are investigated.By studying the maximal good subspaces,we determine the dual Lie coalgebras of the twisted SchrSdinger-Virasoro...In this paper,the structures of dual Lie bialgebras of twisted SchrSdinger-Virasoro type are investigated.By studying the maximal good subspaces,we determine the dual Lie coalgebras of the twisted SchrSdinger-Virasoro algebras.Then based on this,we construct the dual Lie bialgebra structures of this type.As by-products,four new infinite dimensional Lie algebras are obtained.展开更多
By the R-matrix of orthosymplectic quantum superalgebra Uq （osp（2l＋ 1｜2n）） in the vector representation, we establish the corresponding quantum Hopf superalgebra OSPq（2l＋ 1｜2n）. Furthermore, it is shown that...By the R-matrix of orthosymplectic quantum superalgebra Uq （osp（2l＋ 1｜2n）） in the vector representation, we establish the corresponding quantum Hopf superalgebra OSPq（2l＋ 1｜2n）. Furthermore, it is shown that OSPq（2l＋ 1｜2n） is coquasitriangular. Keywords R-matrix, super bialgebra, quantum function superalgebras展开更多
Let A be a bialgebra, R ∈ A A be a strong “cocycle”. It will be shown that the monoidal category _Auhas a braided monoidal subcategory and several equivalent conditions for (A, R ) to be a quasitriangular bialgebra...Let A be a bialgebra, R ∈ A A be a strong “cocycle”. It will be shown that the monoidal category _Auhas a braided monoidal subcategory and several equivalent conditions for (A, R ) to be a quasitriangular bialgebra will be given. Furthermore, it will be shown that A contains a finite dimensional subbialgebra which is a quasitriangular Hopf algebra if R is a YB-operator.展开更多
In this paper, we obtain the following main theorem for a free quadratic bialgebra J: (a) For p≠0, J is a pointed cosemisimple coalgebra. For p=0, J is a hyperalgebra. (b) For p≠0 and q≠0, J has antipode S iff p...In this paper, we obtain the following main theorem for a free quadratic bialgebra J: (a) For p≠0, J is a pointed cosemisimple coalgebra. For p=0, J is a hyperalgebra. (b) For p≠0 and q≠0, J has antipode S iff p·q+2=0 and S(x)=x. Forp=0 or q=0, J has antipode and S(x)=-x. (c) All left J*-modules are rational. Also, we give some applications in homological theory and algebraic K-theory.展开更多
We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the ...We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first eohomology group H1 （W, W × W） is trivial,展开更多
3-Lie algebras have close relationships with many important fields in mathemat- ics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The ...3-Lie algebras have close relationships with many important fields in mathemat- ics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras and the relationships with 3-Lie algebras are studied. And the classification of 4-dimensional 3-Lie coalgebras and 3-dimensional 3-Lie bialgebras over an algebraically closed field of char- acteristic zero are provided.展开更多
基金National Natural Science Foundation of P.R.China No.10571153Post-Doctoral Program of P.R.China,No.2005037713+1 种基金Post-Doctoral Program of Jiangsu Province of China No.0203003403National Science Foundation of Jiangsu Province of China
文摘This paper gives a sufficient and necessary condition for a bialgebra to be a Long bialgebra, and proves a braided product to be a Long bialgebra under some conditions. It also gives a direct sum decomposition of quantum Yang-Baxter modules over Long bialgebras.
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This work was supported by the National Natural Science Foundation of China (Grant No.10471091)"One Hundred Talents Program"from University of Science and Technology of China
文摘We first prove that,for any generalized Hamiltonian type Lie algebra H,the first co- homology group H^1(H,H(?)H) is trivial.We then show that all Lie bialgebra structures on H are triangular.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12171129,11871421)the Zhejiang Provincial Natural Science Foundation of China(Grant No.LY20A010022)the Scientific Research Foundation of Hangzhou Normal University(Grant No.2019QDL012)。
文摘Gel'fand-Dorfman bialgebra,which is both a Lie algebra and a Novikov algebra with some compatibility condition,appeared in the study of Hamiltonian pairs in completely integrable systems.They also emerged in the description of a class special Lie conformal algebras called quadratic Lie conformal algebras.In this paper,we investigate the extending structures problem for Gel'fand-Dorfman bialgebras,which is equivalent to some extending structures problem of quadratic Lie conformal algebras.Explicitly,given a Gel'fand-Dorfman bialgebra(A,o,[.,.]),this problem is how to describe and classify all Gel'fand-Dorfman bialgebra structures on a vector space E(A⊂E)such that(A,o,[.,.])is a subalgebra of E up to an isomorphism whose restriction on A is the identity map.Motivated by the theories of extending structures for Lie algebras and Novikov algebras,we construct an object gH2(V,A)to answer the extending structures problem by introducing the notion of a unified product for Gel'fand-Dorfman bialgebras,where V is a complement of A in E.In particular,we investigate the special case when dim(V)=1 in detail.
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.
文摘Jordan D-bialgebras were introduced by Zhelyabin.In this paper,we use a new approach to study Jordan D-bialgebras by a new notion of the dual representation of the regular representation of a Jordan algebra.Motivated by the essential connection between Lie bialgebras and Manin triples,we give an explicit proof of the equivalence between Jordan D-bialgebras and a class of special Jordan-Manin triples called double constructions of pseudo-euclidean Jordan algebras.We also show that a Jordan D-bialgebra leads to the Jordan Yang-Baxter equation under the coboundary condition and an antisymmetric nondegenerate solution of the Jordan Yang-Baxter equation corresponds to an antisymmetric bilinear form,which we call a Jordan symplectic form on Jordan algebras.Furthermore,there exists a new algebra structure called pre-Jordan algebra on Jordan algebras with a Jordan symplectic form.
基金The authors would like to thank the referees for pointing out several errors and hnprecisions and Professors Yiming Long, Yongbin Ruan and Gang Tian for their helpful communications and supports during these years. H. -L. Her was partially supported by the National Natural Science Foundation of China （Grant No. 10901084） S. Sun was partially supported by the National Natural Science Foundation of China （Grant Nos. 10731080, 11131004）, PHR201106118, and the Institute of Mathematics and Interdisciplinary Science at Capital Normal University all authors are also partially supported by the National Natural Science Foundation of China （Grant No. 11271269）.
文摘Let M be an exact symplectic manifold with contact type boundary such that cl（M） = O. Motivated by noncommutative symplectic geometry and string topology, we show that the cyclic cohomology of the Fukaya category of M has an involutive Lie bialgebra structure.
基金Supported by NSF grant of China and NSF grant of Shandong Province(Grant Nos.11431010,11671056,ZR2013AL013 and ZR2014AL001)
文摘In the present paper, we investigate the dual Lie coalgebras of the centerless W(2,2) algebra by studying the maximal good subspaces. Based on this, we construct the dual Lie bialgebra structures of the centerless W(2,2) Lie bialgebra. As by-products, four new infinite dimensional Lie algebras are obtained.
文摘It is shown that the dual bialgebra of any quasitriangular bialgebra is braided, and the dual bialgebra of some braided bialgebra is quasitriangular.Also it is proved that every nondegenerate finite dimensional braided (dually, quasitriangular) bialgebra has an antipode.
文摘The present paper is a continuation of [1], where we considered braided infinitesimal Hopf algebras (i.e., infinitesimal Hopf algebras in the Yetter-Drin feld category for any Hopf algebra H), and constructed their Drinfeld double as a generalization of Aguiar’s result. In this paper we mainly investigate the necessary and sufficient condition for a braided infinitesimal bialgebra to be a braided Lie bialgebra (i.e., a Lie bialgebra in the category ).
文摘In this paper,the structures of dual Lie bialgebras of twisted SchrSdinger-Virasoro type are investigated.By studying the maximal good subspaces,we determine the dual Lie coalgebras of the twisted SchrSdinger-Virasoro algebras.Then based on this,we construct the dual Lie bialgebra structures of this type.As by-products,four new infinite dimensional Lie algebras are obtained.
基金Supported by National Natural Science Foundation of China （Grant Nos. 10671016, 10771014） and Foundation of Selected Excellent Science and Technology Activity for Returned Scholars of Beijing
文摘By the R-matrix of orthosymplectic quantum superalgebra Uq （osp（2l＋ 1｜2n）） in the vector representation, we establish the corresponding quantum Hopf superalgebra OSPq（2l＋ 1｜2n）. Furthermore, it is shown that OSPq（2l＋ 1｜2n） is coquasitriangular. Keywords R-matrix, super bialgebra, quantum function superalgebras
文摘Let A be a bialgebra, R ∈ A A be a strong “cocycle”. It will be shown that the monoidal category _Auhas a braided monoidal subcategory and several equivalent conditions for (A, R ) to be a quasitriangular bialgebra will be given. Furthermore, it will be shown that A contains a finite dimensional subbialgebra which is a quasitriangular Hopf algebra if R is a YB-operator.
基金Partially supported by the National Natural Science Foundation of China.
文摘In this paper, we obtain the following main theorem for a free quadratic bialgebra J: (a) For p≠0, J is a pointed cosemisimple coalgebra. For p=0, J is a hyperalgebra. (b) For p≠0 and q≠0, J has antipode S iff p·q+2=0 and S(x)=x. Forp=0 or q=0, J has antipode and S(x)=-x. (c) All left J*-modules are rational. Also, we give some applications in homological theory and algebraic K-theory.
文摘We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first eohomology group H1 （W, W × W） is trivial,
基金partially supported by NSF(11371245)of ChinaNSF(A2010000194)of Hebei Province
文摘3-Lie algebras have close relationships with many important fields in mathemat- ics and mathematical physics. This article concerns 3-Lie algebras. The concepts of 3-Lie coalgebras and 3-Lie bialgebras are given. The structures of such categories of algebras and the relationships with 3-Lie algebras are studied. And the classification of 4-dimensional 3-Lie coalgebras and 3-dimensional 3-Lie bialgebras over an algebraically closed field of char- acteristic zero are provided.
