In this paper, we consider Lie triple systems with derivations. A pair consisting of a Lie triple system and a distinguished derivation is called a LietsDer pair. We define a cohomology theory for LietsDer pair with c...In this paper, we consider Lie triple systems with derivations. A pair consisting of a Lie triple system and a distinguished derivation is called a LietsDer pair. We define a cohomology theory for LietsDer pair with coefficients in a representation. We study central extensions of a LietsDer pair. In the next, we generalize the formal deformation theory to LietsDer pairs in which we deform both the Lie triple system bracket and the distinguished derivation. It is governed by the cohomology of LietsDer pair with coefficients in itself.展开更多
In this article, we discuss some properties of a supersymmetric invariant bilinear form on Lie supertriple systems. In particular, a supersymmetric invariant bilinear form on Lie supertriple systems can be extended to...In this article, we discuss some properties of a supersymmetric invariant bilinear form on Lie supertriple systems. In particular, a supersymmetric invariant bilinear form on Lie supertriple systems can be extended to its standard imbedding Lie superalgebras. Furthermore, we generalize Garland＇s theory of universal central extensions for Lie supertriple systems following the classical one for Lie superalgebras. We solve the problems of lifting automorphisms and lifting derivations.展开更多
Constructing Lie superalgebras from Lie algebras is an important subject for studying Lie superalgebras.In this paper,all the superizations are presented for finite-dimensional trivial central extensions of a class of...Constructing Lie superalgebras from Lie algebras is an important subject for studying Lie superalgebras.In this paper,all the superizations are presented for finite-dimensional trivial central extensions of a class of 3-dimensional complex or real Lie algebras.In particular,up to isomorphism,superizations are determined for 1-dimensional trivial central extensions of these Lie algebras by considering orbits of an action of G(g0)on Symad(g0),where Symad(g0)consists of go-invariant symmetric bilinear mappings and G(g0)={(σ,τ)∈Aut(g0)×GL(g0)|[σ(x),τ(y)]=τ([x,y]),x,y∈g0}.展开更多
We completely determine the universal central extension of the generalized orthosymplectic Lie superalgebra ospm 12n（R,-） that is coordinatized by an arbitrary unital associative superalgebra （R,-） with superinvol...We completely determine the universal central extension of the generalized orthosymplectic Lie superalgebra ospm 12n（R,-） that is coordinatized by an arbitrary unital associative superalgebra （R,-） with superinvolution. As a result, an identification between the second homology group of the Lie superalgebra ospm｜2n （R,-） and the first skew-dihedral homology group of the associative superalgebra （R,-） with superin-volution is created for positive integers m and n with （m, n）≠ （1, 1） and （m, n）≠（2, 1）. The second homology groups of the Lie superalgebras ospm1｜2（R,-） and ospm｜2n （R,-） are also characterized explicitly.展开更多
In this article,the authors obtain some results concerning derivations of finitely generated Lie color algebras and discuss the relation between skew derivation space SkDer（L）and central extension H^2（L,F）on some ...In this article,the authors obtain some results concerning derivations of finitely generated Lie color algebras and discuss the relation between skew derivation space SkDer（L）and central extension H^2（L,F）on some Lie color algebras.Meanwhile,they generalize the notion of double extension to quadratic Lie color algebras,a sufficient condition for a quadratic Lie color algebra to be a double extension and further properties are given.展开更多
Assume that N, F and G are groups. If there exsits N, a normal subgroup of G such that N ≌ G and GIN ≌ F, then G is called a central extension of N by F. In this paper, the central extension of N by a minimal non-ab...Assume that N, F and G are groups. If there exsits N, a normal subgroup of G such that N ≌ G and GIN ≌ F, then G is called a central extension of N by F. In this paper, the central extension of N by a minimal non-abelian p-group is determined, where N is an elementary abelian p-group of order p3. Together with our previous work, all central extensions of N by a minimal non-abelian p-group is determined, where N is an elementary abelian p-group.展开更多
In this paper, we study a class of subalgebras of the Lie algebra of vector fields on n-dimensional torus, which are called the Triangular derivation Lie algebra. We give the structure and the central extension of Tri...In this paper, we study a class of subalgebras of the Lie algebra of vector fields on n-dimensional torus, which are called the Triangular derivation Lie algebra. We give the structure and the central extension of Triangular derivation Lie algebra.展开更多
The central extension of the (1+1)-dimensional Poincaré algebra by including fermionic charges which obey not supersymmetric algebra, but a special graded algebra containing in the right hand side a central eleme...The central extension of the (1+1)-dimensional Poincaré algebra by including fermionic charges which obey not supersymmetric algebra, but a special graded algebra containing in the right hand side a central element only is obtained. The corresponding theory being the fermionic extension of the lineal gravity is proposed. We considered the algebra of generators, the field transformations and found Lagrangian and equation of motion, then we derived the Casimir operator and obtained the con- stant black hole mass.展开更多
In this paper, we give a complete classification of eight dimensional nilpotent Lie algebras with four-dimensional center by using the method of Skjelbred and Sund.
Hom-Leibniz algebra is a natural generalization of Leibniz algebras and Hom-Lie algebras. In this paper, we develop some structure theory （such as （co）homology groups, universal central extensions） of Hom-Leibniz ...Hom-Leibniz algebra is a natural generalization of Leibniz algebras and Hom-Lie algebras. In this paper, we develop some structure theory （such as （co）homology groups, universal central extensions） of Hom-Leibniz algebras based on some works of Loday and Pirashvili.展开更多
We study the structure of the generalized 2-dim affine-Virasoro algebra, and describe its automorphism group. Furthermore, we also determine the irreducibility of a Verma module over the generalized 2-dim affine-Viras...We study the structure of the generalized 2-dim affine-Virasoro algebra, and describe its automorphism group. Furthermore, we also determine the irreducibility of a Verma module over the generalized 2-dim affine-Virasoro algebra.展开更多
In this paper, we classify finite 2-groups all of whose nonnormal subgroups have orders at most 2^3. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of P...In this paper, we classify finite 2-groups all of whose nonnormal subgroups have orders at most 2^3. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.展开更多
The q-deformation of W（2, 2） Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the ...The q-deformation of W（2, 2） Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the q-deformation of W（2, 2） Lie algebra are completely determined. Finally, the 1-dimensional central extension of the q-deformed W（2, 2） Lie algebra is studied, which turns out to be coincided with the conventional W（2, 2） Lie algebra in the q → 1 limit.展开更多
The automorphism group of G is determined, where G is a nonabelian p-group given by a central extension as 1→Zpm→G→Zp×…×Zp→1 such that its derived subgroup has order p.
Let g be a(twisted or untwisted)affine Kac-Moody algebra,and μ be a diagram automorphism of g.In this paper,we give an explicit realization for the universal central extensionˆg[μ]of the twisted loop algebra of g wi...Let g be a(twisted or untwisted)affine Kac-Moody algebra,and μ be a diagram automorphism of g.In this paper,we give an explicit realization for the universal central extensionˆg[μ]of the twisted loop algebra of g with respect toμ,which provides a Moody-Rao-Yokonuma presentation for the algebraˆg[μ]whenμis non-transitive,and the presentation is indeed related to the quantization of twisted toroidal Lie algebras.展开更多
Hom-Malcev superalgebras can be considered as a deformation of Malcev superalgebras. We give the definition of Hom-Malcev superalgebras. Moreover, we characterize the Hom-Malcev operator and the representation of Hom-...Hom-Malcev superalgebras can be considered as a deformation of Malcev superalgebras. We give the definition of Hom-Malcev superalgebras. Moreover, we characterize the Hom-Malcev operator and the representation of Hom-Malcev superalgebras. Finally, we study the central extension and the double extension of Hom-Malcev superalgebras.展开更多
The loop-Witt algebra is the Lie algebra of the tensor product of the Witt algebra and the Laurent polynomial algebra. In this paper we study the universal central extension, derivations and automorphism group for the...The loop-Witt algebra is the Lie algebra of the tensor product of the Witt algebra and the Laurent polynomial algebra. In this paper we study the universal central extension, derivations and automorphism group for the loop-Witt algebra.展开更多
Toroidal Leibniz algebras are the universal central extensions of the iterated loop algebras g×C[t1^±1,...,tv^±1] in the category of Leibniz algebras. In this paper, some properties and representations ...Toroidal Leibniz algebras are the universal central extensions of the iterated loop algebras g×C[t1^±1,...,tv^±1] in the category of Leibniz algebras. In this paper, some properties and representations of toroidal Leibniz algebras are studied. Some general theories of central extensions of Leibniz algebras are also obtained.展开更多
Assume G is a group of order 2^(n),n≥5.Let s_(k)(G)denote the number of subgroups of order 2^(k) of G.We classify finite 2-groups G with s k(G)≤2^(4),where 1≤k≤n.
In this paper, we study an infinite-dimensional Lie algebra Bq, called the q-analog Klein bottle Lie algebra. We show that Bq is a finitely generated simple Lie algebra with a unique （up to scalars） symmetric invari...In this paper, we study an infinite-dimensional Lie algebra Bq, called the q-analog Klein bottle Lie algebra. We show that Bq is a finitely generated simple Lie algebra with a unique （up to scalars） symmetric invariant bilinear form. The derivation algebra and the universal central extension of Bq are also determined.展开更多
基金Supported by the National Natural Science Foundation of China (Grant No. 12161013)the General Project of Guizhou University of Finance and Economics (Grant No. 2021KYYB16)。
文摘In this paper, we consider Lie triple systems with derivations. A pair consisting of a Lie triple system and a distinguished derivation is called a LietsDer pair. We define a cohomology theory for LietsDer pair with coefficients in a representation. We study central extensions of a LietsDer pair. In the next, we generalize the formal deformation theory to LietsDer pairs in which we deform both the Lie triple system bracket and the distinguished derivation. It is governed by the cohomology of LietsDer pair with coefficients in itself.
基金supported by the NSFC(10871057)NSFJL(20130101068JC)supported by Fundamental Research Funds for the Central Universities of China and SRFHLJED(12521157)
文摘In this article, we discuss some properties of a supersymmetric invariant bilinear form on Lie supertriple systems. In particular, a supersymmetric invariant bilinear form on Lie supertriple systems can be extended to its standard imbedding Lie superalgebras. Furthermore, we generalize Garland＇s theory of universal central extensions for Lie supertriple systems following the classical one for Lie superalgebras. We solve the problems of lifting automorphisms and lifting derivations.
基金S.Q.Zhao was supported by the NSF of Hainan Province(No.121MS0784)Postgraduate Innovation and Scientific Research Topic of the School of Mathematical Statistics of Hainan Normal University(No.styc202201)+1 种基金W.D.Liu was supported by the NSF of China(No.12061029)the NSF of Hainan Province(No.120RC587).
文摘Constructing Lie superalgebras from Lie algebras is an important subject for studying Lie superalgebras.In this paper,all the superizations are presented for finite-dimensional trivial central extensions of a class of 3-dimensional complex or real Lie algebras.In particular,up to isomorphism,superizations are determined for 1-dimensional trivial central extensions of these Lie algebras by considering orbits of an action of G(g0)on Symad(g0),where Symad(g0)consists of go-invariant symmetric bilinear mappings and G(g0)={(σ,τ)∈Aut(g0)×GL(g0)|[σ(x),τ(y)]=τ([x,y]),x,y∈g0}.
基金supported by National Natural Science Foundation of China (Grant No. 11501213)the China Postdoctoral Science Foundation (Grant No. 2015M570705)+2 种基金the Fundamental Research Funds for the Central Universities (Grant No. 2015ZM085)the China Postdoctoral Science Foundation (Grant No. 2015M571928)the Fundamental Research Funds for the Central Universities
文摘We completely determine the universal central extension of the generalized orthosymplectic Lie superalgebra ospm 12n（R,-） that is coordinatized by an arbitrary unital associative superalgebra （R,-） with superinvolution. As a result, an identification between the second homology group of the Lie superalgebra ospm｜2n （R,-） and the first skew-dihedral homology group of the associative superalgebra （R,-） with superin-volution is created for positive integers m and n with （m, n）≠ （1, 1） and （m, n）≠（2, 1）. The second homology groups of the Lie superalgebras ospm1｜2（R,-） and ospm｜2n （R,-） are also characterized explicitly.
基金National Natural Science Foundation of China(10271076)
文摘In this article,the authors obtain some results concerning derivations of finitely generated Lie color algebras and discuss the relation between skew derivation space SkDer（L）and central extension H^2（L,F）on some Lie color algebras.Meanwhile,they generalize the notion of double extension to quadratic Lie color algebras,a sufficient condition for a quadratic Lie color algebra to be a double extension and further properties are given.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1137123211471198)
文摘Assume that N, F and G are groups. If there exsits N, a normal subgroup of G such that N ≌ G and GIN ≌ F, then G is called a central extension of N by F. In this paper, the central extension of N by a minimal non-abelian p-group is determined, where N is an elementary abelian p-group of order p3. Together with our previous work, all central extensions of N by a minimal non-abelian p-group is determined, where N is an elementary abelian p-group.
基金Supported by the National Natural Science Foundation of China (Grant No. 11171294)the Natural Science Foundation of Heilongjiang Province (Grant No. A201013)the Fund of Heilongjiang Education Committee(Grant No. 11541268)
文摘In this paper, we study a class of subalgebras of the Lie algebra of vector fields on n-dimensional torus, which are called the Triangular derivation Lie algebra. We give the structure and the central extension of Triangular derivation Lie algebra.
文摘The central extension of the (1+1)-dimensional Poincaré algebra by including fermionic charges which obey not supersymmetric algebra, but a special graded algebra containing in the right hand side a central element only is obtained. The corresponding theory being the fermionic extension of the lineal gravity is proposed. We considered the algebra of generators, the field transformations and found Lagrangian and equation of motion, then we derived the Casimir operator and obtained the con- stant black hole mass.
基金Supported by the National Natural Science Foundation of China (Grant No.J1103110)
文摘In this paper, we give a complete classification of eight dimensional nilpotent Lie algebras with four-dimensional center by using the method of Skjelbred and Sund.
基金Supported by National Natural Science Foundation of China （Grant Nos. 10825101, 11047030） and Natural Science Foundation of He＇nan Provincial Education Department （Grant No. 2010Bl10003）
文摘Hom-Leibniz algebra is a natural generalization of Leibniz algebras and Hom-Lie algebras. In this paper, we develop some structure theory （such as （co）homology groups, universal central extensions） of Hom-Leibniz algebras based on some works of Loday and Pirashvili.
文摘We study the structure of the generalized 2-dim affine-Virasoro algebra, and describe its automorphism group. Furthermore, we also determine the irreducibility of a Verma module over the generalized 2-dim affine-Virasoro algebra.
文摘In this paper, we classify finite 2-groups all of whose nonnormal subgroups have orders at most 2^3. Together with a known result, we completely solved Problem 2279 proposed by Y. Berkovich and Z. Janko in Groups of Prime Power Order, Vol. 3.
基金Supported by National Natural Science Foundation of China (Grant No. 10825101)
文摘The q-deformation of W（2, 2） Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the q-deformation of W（2, 2） Lie algebra are completely determined. Finally, the 1-dimensional central extension of the q-deformed W（2, 2） Lie algebra is studied, which turns out to be coincided with the conventional W（2, 2） Lie algebra in the q → 1 limit.
基金Project supported by NSFC （11371124, 11301150） and the Natural Science Foundation of Henan Province of China （142300410134, 162300410066）.
文摘The automorphism group of G is determined, where G is a nonabelian p-group given by a central extension as 1→Zpm→G→Zp×…×Zp→1 such that its derived subgroup has order p.
基金supported by National Natural Science Foundation of China(Grant Nos.11531004 and 11701183)the Fundamental Research Funds for the Central Universities(Grant No.20720190069)the Simons Foundation(Grant No.198129)。
文摘Let g be a(twisted or untwisted)affine Kac-Moody algebra,and μ be a diagram automorphism of g.In this paper,we give an explicit realization for the universal central extensionˆg[μ]of the twisted loop algebra of g with respect toμ,which provides a Moody-Rao-Yokonuma presentation for the algebraˆg[μ]whenμis non-transitive,and the presentation is indeed related to the quantization of twisted toroidal Lie algebras.
基金Acknowledgements The authors would like to thank the referees for their helpful comments to improve the paper. This work was supported in part by the Research Fund for the Doctoral Program of Higher Education of China （No. 201101647） and the Natural Science Foundation of Jilin Province （No. 20130101068）.
文摘Hom-Malcev superalgebras can be considered as a deformation of Malcev superalgebras. We give the definition of Hom-Malcev superalgebras. Moreover, we characterize the Hom-Malcev operator and the representation of Hom-Malcev superalgebras. Finally, we study the central extension and the double extension of Hom-Malcev superalgebras.
基金Supported in part by National Natural Science Foundation of China (Grant No. 11171294)Natural Science Foundation of Heilongjiang Province of China (Grant No. A201013)+2 种基金Science Fundation for Distinguished Young Scholars of Heilongjiang Province of China (Grant No. JC201004)Postdoctoral Scientific Research Foundation of Heilongjiang Province (Grant No. LBH-Q08026)the fund of Heilongjiang Education Committee (Grant No. 11541268)
文摘The loop-Witt algebra is the Lie algebra of the tensor product of the Witt algebra and the Laurent polynomial algebra. In this paper we study the universal central extension, derivations and automorphism group for the loop-Witt algebra.
基金the NNSF (Grants 10671027,10271076,10701019)the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No.06KJBll0003)+1 种基金Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT)the Shanghai Priority Academic Discipline from the SMEC
文摘Toroidal Leibniz algebras are the universal central extensions of the iterated loop algebras g×C[t1^±1,...,tv^±1] in the category of Leibniz algebras. In this paper, some properties and representations of toroidal Leibniz algebras are studied. Some general theories of central extensions of Leibniz algebras are also obtained.
文摘Assume G is a group of order 2^(n),n≥5.Let s_(k)(G)denote the number of subgroups of order 2^(k) of G.We classify finite 2-groups G with s k(G)≤2^(4),where 1≤k≤n.
基金The first author was supported in part by the NSFC （10931006, 10871125） and the Innovation Program of Shanghai Municipal Education Commission （11ZZ18）. The second author was supported by the NSFC （11326060）. The third author was supported in part by the NSFC （11101285, 11026042, 11071068）, the Shanghai Natural Science Foundation （11ZR1425900）, the Innovation Program of Shanghai Municipal Education Commission （11YZ85）, the Academic Discipline Project of Shanghai Normal University （DZL803） and ZJNSF （Y6100148）.
文摘In this paper, we study an infinite-dimensional Lie algebra Bq, called the q-analog Klein bottle Lie algebra. We show that Bq is a finitely generated simple Lie algebra with a unique （up to scalars） symmetric invariant bilinear form. The derivation algebra and the universal central extension of Bq are also determined.
