First, we present semisimple properties of twisted products by means of constructing an algebra isomorphism between twisted products and crossed products, and point out that there exist some relations among braided bi...First, we present semisimple properties of twisted products by means of constructing an algebra isomorphism between twisted products and crossed products, and point out that there exist some relations among braided bialgebras, paired bialgebras and Yang-Baxter coalgebras. Furthermore, we give an example to illustrate these relations by using Sweedler's 4-dimensional Hopf algebra. Finally, from starting off with Yang-Baxter coalgebras, we can construct some quadratic bialgebras such that they are braided bialgebras.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
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
A Lie 2-bialgebra is a Lie 2-algebra equipped with a compatible Lie 2-coalgebra structure. In this paper, we give another equivalent description for Lie2-bialgebras by using the structure maps and compatibility condit...A Lie 2-bialgebra is a Lie 2-algebra equipped with a compatible Lie 2-coalgebra structure. In this paper, we give another equivalent description for Lie2-bialgebras by using the structure maps and compatibility conditions. We can use this method to check whether a 2-term direct sum of vector spaces is a Lie 2-bialgebra easily.展开更多
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
In a recent article by Liu,Pei,and Zhu,Lie bialgebra structures on the twisted Heisenberg-Virasoro algebra were determined. By disposing the indexing set, the generalized Heisenberg-Virasoro algebra was considered. It...In a recent article by Liu,Pei,and Zhu,Lie bialgebra structures on the twisted Heisenberg-Virasoro algebra were determined. By disposing the indexing set, the generalized Heisenberg-Virasoro algebra was considered. It is proved that all Lie bialgebra structures on centerless generalized Heisenberg-Virasoro algebra L are coboundary triangular by proving that the first cohomology group H1 (L,V) =0.展开更多
In this article, Lie super-bialgebra structures on generalized super-Virasoro algebras/: are considered. It is proved that all such Lie super-bialgebras are coboundary triangular Lie super-bialgebras if and only if H...In this article, Lie super-bialgebra structures on generalized super-Virasoro algebras/: are considered. It is proved that all such Lie super-bialgebras are coboundary triangular Lie super-bialgebras if and only if Hi( ) = 0.展开更多
文摘First, we present semisimple properties of twisted products by means of constructing an algebra isomorphism between twisted products and crossed products, and point out that there exist some relations among braided bialgebras, paired bialgebras and Yang-Baxter coalgebras. Furthermore, we give an example to illustrate these relations by using Sweedler's 4-dimensional Hopf algebra. Finally, from starting off with Yang-Baxter coalgebras, we can construct some quadratic bialgebras such that they are braided bialgebras.
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 ).
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.
文摘A Lie 2-bialgebra is a Lie 2-algebra equipped with a compatible Lie 2-coalgebra structure. In this paper, we give another equivalent description for Lie2-bialgebras by using the structure maps and compatibility conditions. We can use this method to check whether a 2-term direct sum of vector spaces is a Lie 2-bialgebra easily.
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National Natural Science Foundations of China(No.11001046,No.11201305)the Fundamental Research Funds for the Central Universities+1 种基金Foundation of Outstanding Young Teachers of Donghua University,ChinaInnovation Project of Shanghai Education Committee,China(No.12YZ081)
文摘In a recent article by Liu,Pei,and Zhu,Lie bialgebra structures on the twisted Heisenberg-Virasoro algebra were determined. By disposing the indexing set, the generalized Heisenberg-Virasoro algebra was considered. It is proved that all Lie bialgebra structures on centerless generalized Heisenberg-Virasoro algebra L are coboundary triangular by proving that the first cohomology group H1 (L,V) =0.
基金Supported by the Foundation of Shanghai Education Committee (06FZ029)NSF of China (10471091)"One Hundred Program" from University of Science and Technology of China
文摘In this article, Lie super-bialgebra structures on generalized super-Virasoro algebras/: are considered. It is proved that all such Lie super-bialgebras are coboundary triangular Lie super-bialgebras if and only if Hi( ) = 0.