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
A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket. We construct a bialgebra theory of compatible Lie Mgebras as an analogue of a piiLie bialge...A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket. We construct a bialgebra theory of compatible Lie Mgebras as an analogue of a piiLie bialgebra. They can also be regarded as a ＂compatible version＂ of Lie bialgebras, that is, a pair of Lie biaJgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra. Many properties of compatible Lie bialgebras as the ＂compatible version＂ of the corresponding properties of Lie biaJgebras are presented. In particular, there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang-Baxter equation in compatible Lie algebras as a combination of two classical Yang-Baxter equations in lAe algebras. FUrthermore, a notion of compatible pre-Lie Mgebra is introduced with an interpretation of its close relation with the classical Yang-Baxter equation in compatible Lie a/gebras which leads to a construction of the solutions of the latter. As a byproduct, the compatible Lie bialgebras fit into the framework to construct non-constant solutions of the classical Yang-Baxter equation given by Golubchik and Sokolov.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
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
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, we introduce 3-Horn-Lie bialgebras whose compatibility conditions between the multiplication and comultiplication are given by local cocycle conditions. We study a twisted 3-ary version of the Yang-Baxt...In this paper, we introduce 3-Horn-Lie bialgebras whose compatibility conditions between the multiplication and comultiplication are given by local cocycle conditions. We study a twisted 3-ary version of the Yang-Baxter Equation, called the 3-Lie classical Hom- Yang-Baxter Equation （3-Lie CHYBE）, which is a general form of 3-Lie classical Yang-Baxter Equation and prove that the bialgebras induced by the solutions of 3-Lie CHYBE induce the coboundary local cocycle 3-Horn-Lie bialgebras.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
Let B be the Lie algebra of Block type with the basis(L_(α,i)|α,i ∈ Z,i≥-1)and Lie bracket[L_(α,i),L_(β,j)]=((i+1)β-(j+1)a)L_(α+β,i+j).It is known that every Lie bialgebra structure on B is a triangular cobou...Let B be the Lie algebra of Block type with the basis(L_(α,i)|α,i ∈ Z,i≥-1)and Lie bracket[L_(α,i),L_(β,j)]=((i+1)β-(j+1)a)L_(α+β,i+j).It is known that every Lie bialgebra structure on B is a triangular coboundary Lie bialgebra.This paper is devoted to the study of dual Lie bialgebras of Lie bialgebras of Block type.The explicit structures of the dual Lie bialgebras which have uncountable generators are given.At the same time,we obtain some new series of infinite-dimensional Lie algebras.展开更多
Let A be an arbitrary hereditary abelian category. Lu and Peng (2021) defined the semi-derived Ringel-Hall algebra SDH(A) of A and proved that SDH(A) has a natural basis and is isomorphic to the Drinfeld double Ringel...Let A be an arbitrary hereditary abelian category. Lu and Peng (2021) defined the semi-derived Ringel-Hall algebra SDH(A) of A and proved that SDH(A) has a natural basis and is isomorphic to the Drinfeld double Ringel-Hall algebra of A. In this paper, we introduce a coproduct formula on SDH(A) with respect to the basis of SDH(A) and prove that this coproduct is compatible with the product of SDH(A), and thereby the semi-derived Ringel-Hall algebra of A is endowed with a bialgebra structure which is identified with the bialgebra structure of the Drinfeld double Ringel-Hall algebra of A.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 this paper,we first give a direct sum decomposition of Lie comodules,and then accord- ing to the Lie comodule theory,construct some(triangular)Lie bialgebras through Lie coalgebras.
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
With the purpose of providing a categorical treatment of weak multiplier bialgebras （introduced by BShm, Gomez-Torrecillas and Ldpez-Centella in 2015）, an ap- propriate notion of morphism for these algebraic objects...With the purpose of providing a categorical treatment of weak multiplier bialgebras （introduced by BShm, Gomez-Torrecillas and Ldpez-Centella in 2015）, an ap- propriate notion of morphism for these algebraic objects is proposed. This allows us to define a category wmb of （regular） weak multiplier bialgebras （with a right full comultipli- cation）, containing as a full subcategory the category wba of weak bialgebras defined by BShm, Gomez-Torrecillas and Lopez-Centella in 2014. We present a great source of ex- amples of these morphisms proving that, under some assumption, a functor between small categories induces a morphism of this kind between the natural weak multiplier bialgebra structures carried by the linear spans of the arrow sets of the categories. We explore the notion of elements of group-like type in a weak multiplier bialgebra, proposing a definition in the line of the one by the aforementioned authors for weak bialgebras. We show a big number of its properties and provide more general versions of many results known in the context of weak bialgebras. In particular, in analogy with the classical bialgebra setting （where the set of group-like elements is a monoid）, we prove that the set of these elements possesses a structure of category.展开更多
In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are considered. It is proved that, for any Lie algebra W of generalized Witt type, all Lie bialgebras on W are the coboundary tr...In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are considered. It is proved that, for any Lie algebra W of generalized Witt type, all Lie bialgebras on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W, W(?)W) is trivial.展开更多
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
In this paper, Lie bialgebra structures on generalized Virasoro-like algebras are studied. It is proved that all such Lie bialgebras are triangular coboundary.
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.
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.
基金Supported by National Natural Science Foundation of China under Grant Nos.11271202,11221091,11425104Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20120031110022
文摘A compatible Lie algebra is a pair of Lie algebras such that any linear combination of the two Lie brackets is a Lie bracket. We construct a bialgebra theory of compatible Lie Mgebras as an analogue of a piiLie bialgebra. They can also be regarded as a ＂compatible version＂ of Lie bialgebras, that is, a pair of Lie biaJgebras such that any linear combination of the two Lie bialgebras is still a Lie bialgebra. Many properties of compatible Lie bialgebras as the ＂compatible version＂ of the corresponding properties of Lie biaJgebras are presented. In particular, there is a coboundary compatible Lie bialgebra theory with a construction from the classical Yang-Baxter equation in compatible Lie algebras as a combination of two classical Yang-Baxter equations in lAe algebras. FUrthermore, a notion of compatible pre-Lie Mgebra is introduced with an interpretation of its close relation with the classical Yang-Baxter equation in compatible Lie a/gebras which leads to a construction of the solutions of the latter. As a byproduct, the compatible Lie bialgebras fit into the framework to construct non-constant solutions of the classical Yang-Baxter equation given by Golubchik and Sokolov.
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.
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Supported by the National Natural Science Foundation of China(Grant No.11047030)the Science and Technology Program of Henan Province(Grant No.152300410061)
文摘In this paper, we introduce 3-Horn-Lie bialgebras whose compatibility conditions between the multiplication and comultiplication are given by local cocycle conditions. We study a twisted 3-ary version of the Yang-Baxter Equation, called the 3-Lie classical Hom- Yang-Baxter Equation （3-Lie CHYBE）, which is a general form of 3-Lie classical Yang-Baxter Equation and prove that the bialgebras induced by the solutions of 3-Lie CHYBE induce the coboundary local cocycle 3-Horn-Lie bialgebras.
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 Science Foundation of China(12371026,11771122).
文摘Let B be the Lie algebra of Block type with the basis(L_(α,i)|α,i ∈ Z,i≥-1)and Lie bracket[L_(α,i),L_(β,j)]=((i+1)β-(j+1)a)L_(α+β,i+j).It is known that every Lie bialgebra structure on B is a triangular coboundary Lie bialgebra.This paper is devoted to the study of dual Lie bialgebras of Lie bialgebras of Block type.The explicit structures of the dual Lie bialgebras which have uncountable generators are given.At the same time,we obtain some new series of infinite-dimensional Lie algebras.
文摘Let A be an arbitrary hereditary abelian category. Lu and Peng (2021) defined the semi-derived Ringel-Hall algebra SDH(A) of A and proved that SDH(A) has a natural basis and is isomorphic to the Drinfeld double Ringel-Hall algebra of A. In this paper, we introduce a coproduct formula on SDH(A) with respect to the basis of SDH(A) and prove that this coproduct is compatible with the product of SDH(A), and thereby the semi-derived Ringel-Hall algebra of A is endowed with a bialgebra structure which is identified with the bialgebra structure of the Drinfeld double Ringel-Hall algebra of A.
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the Educational Ministry Key Foundation of China(Grant No.108154)the National Natural Science Foundation of China(Grant No.10571153)
文摘In this paper,we first give a direct sum decomposition of Lie comodules,and then accord- ing to the Lie comodule theory,construct some(triangular)Lie bialgebras through Lie coalgebras.
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With the purpose of providing a categorical treatment of weak multiplier bialgebras （introduced by BShm, Gomez-Torrecillas and Ldpez-Centella in 2015）, an ap- propriate notion of morphism for these algebraic objects is proposed. This allows us to define a category wmb of （regular） weak multiplier bialgebras （with a right full comultipli- cation）, containing as a full subcategory the category wba of weak bialgebras defined by BShm, Gomez-Torrecillas and Lopez-Centella in 2014. We present a great source of ex- amples of these morphisms proving that, under some assumption, a functor between small categories induces a morphism of this kind between the natural weak multiplier bialgebra structures carried by the linear spans of the arrow sets of the categories. We explore the notion of elements of group-like type in a weak multiplier bialgebra, proposing a definition in the line of the one by the aforementioned authors for weak bialgebras. We show a big number of its properties and provide more general versions of many results known in the context of weak bialgebras. In particular, in analogy with the classical bialgebra setting （where the set of group-like elements is a monoid）, we prove that the set of these elements possesses a structure of category.
文摘In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are considered. It is proved that, for any Lie algebra W of generalized Witt type, all Lie bialgebras on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W, W(?)W) is trivial.
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Supported by an NSF Grant 10471096 of China,"One Hundred Talents Program"from University of Science and Technology of China and"Trans-Century Training Programme Foundation for the Talents"from National Education Ministry of China
文摘In this paper, Lie bialgebra structures on generalized Virasoro-like algebras are studied. It is proved that all such Lie bialgebras are triangular coboundary.
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.
