Semenov-Tian-Shansky has given the solution of the modified classical Yang-Baxter equation, which was called the modified r-matrix. Relevant studies have been extensive in recent times. In this paper, we introduce the...Semenov-Tian-Shansky has given the solution of the modified classical Yang-Baxter equation, which was called the modified r-matrix. Relevant studies have been extensive in recent times. In this paper, we introduce the concept and representations of modified RotaBaxter Hom-Lie algebras. We develop a cohomology of modified Rota-Baxter Hom-Lie algebras with coefficients in a suitable representation. As applications, we study formal deformations and abelian extensions of modified Rota-Baxter Hom-Lie algebras in terms of second cohomology groups.展开更多
In this paper,we first introduce the notion of relative Rota-Baxter operators of nonzero weight on 3-Hom-Lie algebras and define a cohomology of relative Rota-Baxter operators of nonzero weight on 3-Hom-Lie algebras w...In this paper,we first introduce the notion of relative Rota-Baxter operators of nonzero weight on 3-Hom-Lie algebras and define a cohomology of relative Rota-Baxter operators of nonzero weight on 3-Hom-Lie algebras with coefficients in a suitable representation.Next,we introduce and study 3-Hom-post-Lie-algebras as the underlying structure of relative Rota-Baxter operators of nonzero weight on 3-Hom-Lie algebras.Finally,we investigate relative Rota-Baxter operators of nonzero weight on 3-Hom-Lie algebras induced by Hom-Lie algebras.展开更多
In this paper,we introduce the notion of embedding tensors on 3-Hom-Lie algebras and show that embedding tensors induce naturally 3-Hom-Leibniz algebras.Moreover,the cohomology theory of embedding tensors on 3-Hom-Lie...In this paper,we introduce the notion of embedding tensors on 3-Hom-Lie algebras and show that embedding tensors induce naturally 3-Hom-Leibniz algebras.Moreover,the cohomology theory of embedding tensors on 3-Hom-Lie algebras is defined.As an application,we show that if two linear deformations of an embedding tensor on a 3-Hom-Lie algebra are equivalent,then their infinitesimals belong to the same cohomology class in the first cohomology group.展开更多
The finite-dimensional indecomposable solvable Lie algebras s with Q2n+1as their nilradical are studied and classified, it turns out that the dimension of s is dim Q2n+1+1.Then the Hom-Lie algebra structures on solvab...The finite-dimensional indecomposable solvable Lie algebras s with Q2n+1as their nilradical are studied and classified, it turns out that the dimension of s is dim Q2n+1+1.Then the Hom-Lie algebra structures on solvable Lie algebras s are calculated.展开更多
Let A be a multiplicative Hom-associative algebra and L a multiplicative Hom-Lie algebra together with surjective twisting maps. We show that if A is a sum of two commutative Hom-associative subalgebras, then the comm...Let A be a multiplicative Hom-associative algebra and L a multiplicative Hom-Lie algebra together with surjective twisting maps. We show that if A is a sum of two commutative Hom-associative subalgebras, then the commutator Hom-ideal is nilpotent. Furthermore, we obtain an analogous result for Hom-Lie algebra L extending Kegel＇s Theorem. Finally, we discuss the Hom-Lie ideal structure of a simple Hom-associative algebra A by showing that any non-commutative Hom-Lie ideal of A must contain [A, A].展开更多
Let H be a Hopf algebra and HYD the Yetter- Drinfeld category over H. First, the enveloping algebra of generalized H-Hom-Lie algebra L, i.e., Hom-Lie algebra L H in the category HYD, is constructed. Secondly, it is o...Let H be a Hopf algebra and HYD the Yetter- Drinfeld category over H. First, the enveloping algebra of generalized H-Hom-Lie algebra L, i.e., Hom-Lie algebra L H in the category HYD, is constructed. Secondly, it is obtained that U（L） = T（ L）/L where I is the Hom-ideal of T（L） generated by {ll＇-l_（（-1））·l＇l_0-[l,l＇]｜l,l＇∈L}, and u： L,T（L）/I is the canonical map. Finally, as the applications of the result, the enveloping algebras of generalized H-Lie algebras, i.e., the Lie algebras in the category MyDn and the Hom-Lie algebras in the category of left H-comodules are presented, respectively.展开更多
The purpose of this paper is to define Hochschild type homology of Bihomassociative algebras and Chevalley-Eilenberg type homology of Bihom-Lie algebras with non-trivial coefficients in their bimodules respectively.In...The purpose of this paper is to define Hochschild type homology of Bihomassociative algebras and Chevalley-Eilenberg type homology of Bihom-Lie algebras with non-trivial coefficients in their bimodules respectively.In particular,we give their low order homology in detail.展开更多
On Hom-Lie algebras and superalgebras,we introduce the notions of biderivations and linear commuting maps,and compute them for some typical Hom-Lie algebras and superalgebras,including the q-deformed W(2,2)algebra,the...On Hom-Lie algebras and superalgebras,we introduce the notions of biderivations and linear commuting maps,and compute them for some typical Hom-Lie algebras and superalgebras,including the q-deformed W(2,2)algebra,the q-deformed Witt algebra and superalgebra.展开更多
In this article,we construct free centroid hom-associative algebras and free centroid hom-Lie algebras.We also construct some other relatively free centroid hom-associative algebras by applying the Gr?bner-Shirshov ba...In this article,we construct free centroid hom-associative algebras and free centroid hom-Lie algebras.We also construct some other relatively free centroid hom-associative algebras by applying the Gr?bner-Shirshov basis theory for(unital)centroid hom-associative algebras.Finally,we prove that the"Poincaré-Birkhoff-Witt theorem"holds for certain type of centroid hom-Lie algebras over a field of characteristic 0,namely,every centroid hom-Lie algebra such that the eigenvectors of the mapβlinearly generates the whole algebra can be embedded into its universal enveloping centroid hom-associative algebra,and the linear basis of the universal enveloping algebra does not depend on the multiplication table of the centroid hom-Lie algebra under consideration.展开更多
A Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps.In this paper,we study representations of Bihom-Lie algebras.In particular,derivations,central extensions,deri...A Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps.In this paper,we study representations of Bihom-Lie algebras.In particular,derivations,central extensions,derivation extensions,the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail.展开更多
This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W（2, 2） algebra. We show that the q-deformed W（2, 2） algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspon...This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W（2, 2） algebra. We show that the q-deformed W（2, 2） algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W（2, 2） algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W（2, 2） algebra.展开更多
In this paper,we introduce the notion of a product structure on a 3-Bihom-Lie algebra,which is a Nijenhuis operator with some conditions.We prove that a 3-Bihom-Lie algebra has a product structure if and only if it is...In this paper,we introduce the notion of a product structure on a 3-Bihom-Lie algebra,which is a Nijenhuis operator with some conditions.We prove that a 3-Bihom-Lie algebra has a product structure if and only if it is the direct sum of two vector spaces which are also Bihom-subalgebras.Then we give four special conditions under each of which a 3-Bihom-Lie algebra has a special decomposition.Similarly,we introduce a complex structure on a 3-Bihom-Lie algebra and there are also four types of special complex structures.Finally,we establish the relation between a complex structure and a product structure.展开更多
In this paper,we consider compatible Hom-Lie triple systems.More precisely,compatible Hom-Lie triple systems are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie algebra.We also define a...In this paper,we consider compatible Hom-Lie triple systems.More precisely,compatible Hom-Lie triple systems are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie algebra.We also define a cohomology theory for compatible Hom-Lie triple systems.As applications of cohomology,we study linear deformations and abelian extensions of compatible Hom-Lie triple systems.展开更多
The aim of this article is to introduce the notion of Hom-Lie H-pseudo-superalgebras for any Hopf algebra H. This class of algebras is a natural generalization of the Hom-Lie pseudo-algebras as well as a special case ...The aim of this article is to introduce the notion of Hom-Lie H-pseudo-superalgebras for any Hopf algebra H. This class of algebras is a natural generalization of the Hom-Lie pseudo-algebras as well as a special case of the Hom-Lie superalgebras. We present some construction theorems of Hom-Lie H-pseudo-superalgebras, reformulate the equivalent definition of Hom-Lie H-pseudo-super-algebras, and consider the cohomology theory of Hom-Lie H-pseudo-superalgebras with coefficients in arbitrary Hom-modules as a generalization of Kac’s result.展开更多
In this paper,we first introduce the notion of relative Rota-Baxter operators on Hom-Lie-Yamaguti algebras and give some characteristics of relative Rota-Baxter operators in terms of Nijenhuis operators and graphs.The...In this paper,we first introduce the notion of relative Rota-Baxter operators on Hom-Lie-Yamaguti algebras and give some characteristics of relative Rota-Baxter operators in terms of Nijenhuis operators and graphs.Then,the cohomology theory of relative Rota-Baxter operators on Hom-Lie-Yamaguti algebras is proposed.Finally,the deformation of the relative Rota-Baxter operator is explored by applying the cohomological approach.展开更多
The modifiedλ-differential Lie-Yamaguti algebras are considered,in which a modifiedλ-differential Lie-Yamaguti algebra consisting of a Lie-Yamaguti algebra and a modifiedλ-differential operator.First we introduce t...The modifiedλ-differential Lie-Yamaguti algebras are considered,in which a modifiedλ-differential Lie-Yamaguti algebra consisting of a Lie-Yamaguti algebra and a modifiedλ-differential operator.First we introduce the representation of modifiedλ-differential Lie-Yamaguti algebras.Furthermore,we establish the cohomology of a modifiedλ-differential Lie-Yamaguti algebra with coefficients in a representation.Finally,we investigate the one-parameter formal deformations and Abelian extensions of modifiedλ-differential Lie-Yamaguti algebras using the second cohomology group.展开更多
A left Leibniz algebra equipped with an invariant nondegenerate skew-symmetric bilinear form(i.e.,a skew-symmetric quadratic Leibniz algebra)is constructed.The notion of T^(*)-extension of Lie-Yamaguti algebras is int...A left Leibniz algebra equipped with an invariant nondegenerate skew-symmetric bilinear form(i.e.,a skew-symmetric quadratic Leibniz algebra)is constructed.The notion of T^(*)-extension of Lie-Yamaguti algebras is introduced and it is observed that the trivial extension of a Lie-Yamaguti algebra is a quadratic Lie-Yamaguti algebra.It is proved that every symmetric(resp.,skew-symmetric)quadratic Leibniz algebra induces a quadratic(resp.,symplectic)LieYamaguti algebra.展开更多
A bottleneck algebra is a linearly ordered set(B,≤)with two operations a⊕b=max{a,b}and a⊗b=min{a,b}.A finite nonempty set of vectors of order m over a bottleneck algebra B is said to be 2 B-independent if each vecto...A bottleneck algebra is a linearly ordered set(B,≤)with two operations a⊕b=max{a,b}and a⊗b=min{a,b}.A finite nonempty set of vectors of order m over a bottleneck algebra B is said to be 2 B-independent if each vector of order m over B can be expressed as a linear combination of vectors in this set in at most one way.In 1996,Cechlárováand Plávka posed an open problem:Find a necessary and sufficient condition for a finite nonempty set of vectors of order m over B to be 2 B-independent.In this paper,we derive some necessary and sufficient conditions for a finite nonempty set of vectors of order m over a bounded bottleneck algebra to be 2 B-independent and answer this open problem.展开更多
基金Supported by the Universities Key Laboratory of System Modeling and Data Mining in Guizhou Province(Grant No.2023013)the National Natural Science Foundation of China(Grant No.12161013)the Science and Technology Program of Guizhou Province(Grant No.ZK[2023]025)。
文摘Semenov-Tian-Shansky has given the solution of the modified classical Yang-Baxter equation, which was called the modified r-matrix. Relevant studies have been extensive in recent times. In this paper, we introduce the concept and representations of modified RotaBaxter Hom-Lie algebras. We develop a cohomology of modified Rota-Baxter Hom-Lie algebras with coefficients in a suitable representation. As applications, we study formal deformations and abelian extensions of modified Rota-Baxter Hom-Lie algebras in terms of second cohomology groups.
基金Supported by the National Natural Science Foundation of China(Grant No.12161013)the School-Level Student Research Project of Guizhou University of Finance and Economics(Grant No.2024ZXSY231)。
文摘In this paper,we first introduce the notion of relative Rota-Baxter operators of nonzero weight on 3-Hom-Lie algebras and define a cohomology of relative Rota-Baxter operators of nonzero weight on 3-Hom-Lie algebras with coefficients in a suitable representation.Next,we introduce and study 3-Hom-post-Lie-algebras as the underlying structure of relative Rota-Baxter operators of nonzero weight on 3-Hom-Lie algebras.Finally,we investigate relative Rota-Baxter operators of nonzero weight on 3-Hom-Lie algebras induced by Hom-Lie algebras.
基金Supported by the Scientific Research Foundation for Science&Technology Innovation Talent Team of the Intelligent Computing and Monitoring of Guizhou Province(Grant No.QJJ[2023]063)the Science and Technology Program of Guizhou Province(Grant Nos.ZK[2023]025+4 种基金QKHZC[2023]372ZK[2022]031)the National Natural Science Foundation of China(Grant No.12161013)the Scientific Research Foundation of Guizhou University of Finance and Economics(Grant No.2022KYYB08)the Doctoral Research Start-Up Fund of Guiyang University(Grant No.GYU-KY-2024).
文摘In this paper,we introduce the notion of embedding tensors on 3-Hom-Lie algebras and show that embedding tensors induce naturally 3-Hom-Leibniz algebras.Moreover,the cohomology theory of embedding tensors on 3-Hom-Lie algebras is defined.As an application,we show that if two linear deformations of an embedding tensor on a 3-Hom-Lie algebra are equivalent,then their infinitesimals belong to the same cohomology class in the first cohomology group.
基金Foundation item： Supported by the National Natural Science Foundation of China（11071187） Supported by the Natural Science Foundation of Henan Province（13A110785）
文摘The finite-dimensional indecomposable solvable Lie algebras s with Q2n+1as their nilradical are studied and classified, it turns out that the dimension of s is dim Q2n+1+1.Then the Hom-Lie algebra structures on solvable Lie algebras s are calculated.
基金Supported by the Excellent Young Talents Fund Project of Anhui Province(Grant No.2013SQRL092ZD)the Natural Science Foundation of Anhui Province(Grant Nos.1408085QA06+2 种基金1408085QA08)the Excellent Young Talents Fund Project of Chuzhou University(Grant No.2013RC001)the Research and Innovation Projectfor College Graduates of Jiangsu Province(Grant No.CXLX12-0071)
文摘Let A be a multiplicative Hom-associative algebra and L a multiplicative Hom-Lie algebra together with surjective twisting maps. We show that if A is a sum of two commutative Hom-associative subalgebras, then the commutator Hom-ideal is nilpotent. Furthermore, we obtain an analogous result for Hom-Lie algebra L extending Kegel＇s Theorem. Finally, we discuss the Hom-Lie ideal structure of a simple Hom-associative algebra A by showing that any non-commutative Hom-Lie ideal of A must contain [A, A].
基金The National Natural Science Foundation of China(No.11371088)the Excellent Young Talents Fund of Anhui Province(No.2013SQRL092ZD)+2 种基金the Natural Science Foundation of Higher Education Institutions of Anhui Province(No.KJ2015A294)China Postdoctoral Science Foundation(No.2015M571725)the Excellent Young Talents Fund of Chuzhou University(No.2013RC001)
文摘Let H be a Hopf algebra and HYD the Yetter- Drinfeld category over H. First, the enveloping algebra of generalized H-Hom-Lie algebra L, i.e., Hom-Lie algebra L H in the category HYD, is constructed. Secondly, it is obtained that U（L） = T（ L）/L where I is the Hom-ideal of T（L） generated by {ll＇-l_（（-1））·l＇l_0-[l,l＇]｜l,l＇∈L}, and u： L,T（L）/I is the canonical map. Finally, as the applications of the result, the enveloping algebras of generalized H-Lie algebras, i.e., the Lie algebras in the category MyDn and the Hom-Lie algebras in the category of left H-comodules are presented, respectively.
基金Supported by the National Science Foundation of China(Grant Nos.11047030,11171055).
文摘The purpose of this paper is to define Hochschild type homology of Bihomassociative algebras and Chevalley-Eilenberg type homology of Bihom-Lie algebras with non-trivial coefficients in their bimodules respectively.In particular,we give their low order homology in detail.
基金Supported by National Natural Science Foundation grants of China(Grant No.11301109)。
文摘On Hom-Lie algebras and superalgebras,we introduce the notions of biderivations and linear commuting maps,and compute them for some typical Hom-Lie algebras and superalgebras,including the q-deformed W(2,2)algebra,the q-deformed Witt algebra and superalgebra.
基金the grant of Guangzhou Civil Aviation College(Grant No.22X0430)the RAS Fundamental Research Program(Grant No.FWNF-2022-0002)+2 种基金the NNSF of China(Grant Nos.11571121,12071156)the NNSF of China(Grant No.12101248)the China Postdoctoral Science Foundation(Grant No.2021M691099)。
文摘In this article,we construct free centroid hom-associative algebras and free centroid hom-Lie algebras.We also construct some other relatively free centroid hom-associative algebras by applying the Gr?bner-Shirshov basis theory for(unital)centroid hom-associative algebras.Finally,we prove that the"Poincaré-Birkhoff-Witt theorem"holds for certain type of centroid hom-Lie algebras over a field of characteristic 0,namely,every centroid hom-Lie algebra such that the eigenvectors of the mapβlinearly generates the whole algebra can be embedded into its universal enveloping centroid hom-associative algebra,and the linear basis of the universal enveloping algebra does not depend on the multiplication table of the centroid hom-Lie algebra under consideration.
基金Supported by the National Science Foundation of China(Nos.11047030 and 11771122).
文摘A Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps.In this paper,we study representations of Bihom-Lie algebras.In particular,derivations,central extensions,derivation extensions,the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail.
基金Supported by China Scholarship Council(Grant No.201206125047)China Postdoctoral Science Foundation Funded Project(Grant No.2012M520715)the Fundamental Research Funds for the Central Universities(Grant No.HIT.NSRIF.201462)
文摘This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W（2, 2） algebra. We show that the q-deformed W（2, 2） algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W（2, 2） algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W（2, 2） algebra.
基金Supported by NNSF of China(No.12271085 and No.12071405)supported by Sichuan Science and Technology Program(No.2023NSFSC1287).
文摘In this paper,we introduce the notion of a product structure on a 3-Bihom-Lie algebra,which is a Nijenhuis operator with some conditions.We prove that a 3-Bihom-Lie algebra has a product structure if and only if it is the direct sum of two vector spaces which are also Bihom-subalgebras.Then we give four special conditions under each of which a 3-Bihom-Lie algebra has a special decomposition.Similarly,we introduce a complex structure on a 3-Bihom-Lie algebra and there are also four types of special complex structures.Finally,we establish the relation between a complex structure and a product structure.
基金Supported by the Scientifc Research Foundation for Advanced Talents of GUFE(Grant No.2022YJ007)the Innovation Exploration and Academic Talent Project of GUFE(Grant No.2022XSXMB11)+4 种基金the Science and Technology Program of Guizhou Province(Grant Nos.QKHZC[2023]372QKHJC-[2024]QN081)the Research Foundation for Science&Technology Innovation Team of Guizhou Province(Grant Nos.QJJ[2023]063QJJ[2024]190)the Doctoral Research Start-Up Fundation of Guiyang University(Grant No.GYU-KY-2024)。
文摘In this paper,we consider compatible Hom-Lie triple systems.More precisely,compatible Hom-Lie triple systems are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie algebra.We also define a cohomology theory for compatible Hom-Lie triple systems.As applications of cohomology,we study linear deformations and abelian extensions of compatible Hom-Lie triple systems.
文摘The aim of this article is to introduce the notion of Hom-Lie H-pseudo-superalgebras for any Hopf algebra H. This class of algebras is a natural generalization of the Hom-Lie pseudo-algebras as well as a special case of the Hom-Lie superalgebras. We present some construction theorems of Hom-Lie H-pseudo-superalgebras, reformulate the equivalent definition of Hom-Lie H-pseudo-super-algebras, and consider the cohomology theory of Hom-Lie H-pseudo-superalgebras with coefficients in arbitrary Hom-modules as a generalization of Kac’s result.
基金Supported by the Foundation of Science and Technology of Guizhou Province(Grant No.[2018]1020)the Guizhou University of Finance and Economics introduced talent research projects(2016)the National Natural Science Foundation of China(Grant No.12161013)。
文摘In this paper,we first introduce the notion of relative Rota-Baxter operators on Hom-Lie-Yamaguti algebras and give some characteristics of relative Rota-Baxter operators in terms of Nijenhuis operators and graphs.Then,the cohomology theory of relative Rota-Baxter operators on Hom-Lie-Yamaguti algebras is proposed.Finally,the deformation of the relative Rota-Baxter operator is explored by applying the cohomological approach.
基金National Natural Science Foundation of China(12161013)Research Projects of Guizhou University of Commerce in 2024。
文摘The modifiedλ-differential Lie-Yamaguti algebras are considered,in which a modifiedλ-differential Lie-Yamaguti algebra consisting of a Lie-Yamaguti algebra and a modifiedλ-differential operator.First we introduce the representation of modifiedλ-differential Lie-Yamaguti algebras.Furthermore,we establish the cohomology of a modifiedλ-differential Lie-Yamaguti algebra with coefficients in a representation.Finally,we investigate the one-parameter formal deformations and Abelian extensions of modifiedλ-differential Lie-Yamaguti algebras using the second cohomology group.
文摘A left Leibniz algebra equipped with an invariant nondegenerate skew-symmetric bilinear form(i.e.,a skew-symmetric quadratic Leibniz algebra)is constructed.The notion of T^(*)-extension of Lie-Yamaguti algebras is introduced and it is observed that the trivial extension of a Lie-Yamaguti algebra is a quadratic Lie-Yamaguti algebra.It is proved that every symmetric(resp.,skew-symmetric)quadratic Leibniz algebra induces a quadratic(resp.,symplectic)LieYamaguti algebra.
基金Supported by National Natural Science Foundation of China(Grant Nos.11771004 and 11971111).
文摘A bottleneck algebra is a linearly ordered set(B,≤)with two operations a⊕b=max{a,b}and a⊗b=min{a,b}.A finite nonempty set of vectors of order m over a bottleneck algebra B is said to be 2 B-independent if each vector of order m over B can be expressed as a linear combination of vectors in this set in at most one way.In 1996,Cechlárováand Plávka posed an open problem:Find a necessary and sufficient condition for a finite nonempty set of vectors of order m over B to be 2 B-independent.In this paper,we derive some necessary and sufficient conditions for a finite nonempty set of vectors of order m over a bounded bottleneck algebra to be 2 B-independent and answer this open problem.
