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 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
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
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
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
The aim of this paper is to introduce and study Hom-Gel＇fand-Dorfman super-bialgebras and Hom-Lie conformal superalgebras. In this paper, we provide different ways for constructing Hom- Gel＇fand-Dorfman super-bialge...The aim of this paper is to introduce and study Hom-Gel＇fand-Dorfman super-bialgebras and Hom-Lie conformal superalgebras. In this paper, we provide different ways for constructing Hom- Gel＇fand-Dorfman super-bialgebras. Also, we obtain some infinite-dimensional Hom-Lie superalgebras from affinization of Hom-Gel＇fand-Dorfman super-bialgebras. Finally, we give a general construction of Hom-Lie conformal superalgebras from Hom-Lie superalgebras and establish the equivalence between quadratic Hom-Lie conformal superalgebras and Hom-Gel＇fand-Dorfman super-bialgebras.展开更多
The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie con...The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce α~k-derivations of multiplicative Hom-Lie conformal algebras and study their properties.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 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 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 National Natural Science Foundation grants of China(Grant No.11301109)the Research Fund for the Doctoral Program of Higher Education(Grant No.20132302120042)
文摘The aim of this paper is to introduce and study Hom-Gel＇fand-Dorfman super-bialgebras and Hom-Lie conformal superalgebras. In this paper, we provide different ways for constructing Hom- Gel＇fand-Dorfman super-bialgebras. Also, we obtain some infinite-dimensional Hom-Lie superalgebras from affinization of Hom-Gel＇fand-Dorfman super-bialgebras. Finally, we give a general construction of Hom-Lie conformal superalgebras from Hom-Lie superalgebras and establish the equivalence between quadratic Hom-Lie conformal superalgebras and Hom-Gel＇fand-Dorfman super-bialgebras.
基金supported by National Natural Science Foundation of China (Grant Nos. 11171055, 11471090 and 11301109)Natural Science Foundation of Jilin Province (Grant No. 20170101048JC)
文摘The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce α~k-derivations of multiplicative Hom-Lie conformal algebras and study their properties.
