Leibniz代数上的L-算子和Lb-方程
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摘要
本篇文章我们主要研究了将Lie代数推广为Leibniz代数时,与Lie代数上的O-算子和经典Yang-Baxter方程相对应的算子和代数方程,即Leibniz代数相对于某个的双模(表示)的L-算子和Leibniz代数上的Lb-方程.我们给出了L-算子和Lb-方程的定义并研究了它们之间的关系.同时,由对Lie代数和pre-Lie代数研究启发,我们给出了quasi-pre-Lie代数的定义,研究了Leibniz代数和quasi-pre-Lie代数的关系,证明了quasi-pre-Lie代数给出了其邻接Leibniz代数的一种模结构以及quasi-pre-Lie代数与这种模结构的等价性,并且讨论了quasi-pre-Lie代数与其邻接Leibniz代数上的Lb-方程的关系.另外,我们在研究Leibniz代数的双模(表示)以及对偶双模时还得出了两个附属结果,即由Leibniz代数的双模我们给出了两个Leibuiz代数L_1,L_2形成一个matched pair的充分必要条件;利用对偶双模我们给出Leibniz代数上一种重要的非退化反对称双线性型.
In our paper,we mainly study when Lie algebras are generalized to Leibniz algebras,the operators and the equation which corresponding to O-operators and Classcial Yang-Baxter equation(CYBE) in Lie algebras,that is,the L-operators and Lb-equation on Leibniz algebras.We give the definitions of L-operators and Lb-equation and we also research the relations between them.At the same time,by studying the relations between Lie algebras and pre-Lie algebras,we give the notion of quasi-pre-Lie algebra,and we also study the relations between the quasi-pre-Lie algebra and Lb-equation.Furthermore,we get two subsidiary conclusions when we study the bimodules and dual bimodules of Leibniz algebras:one is we use the bimodule to get a necessary and sufficient condition about matched pair construction by two Leibniz algebras and the other is we use the dual bimodule to get an important nondegenerate skew-symmetric bilinear form on Leibniz algebras.
In our paper,we mainly study when Lie algebras are generalized to Leibniz algebras,the operators and the equation which corresponding to O-operators and Classcial Yang-Baxter equation(CYBE) in Lie algebras,that is,the L-operators and Lb-equation on Leibniz algebras.We give the definitions of L-operators and Lb-equation and we also research the relations between them.At the same time,by studying the relations between Lie algebras and pre-Lie algebras,we give the notion of quasi-pre-Lie algebra,and we also study the relations between the quasi-pre-Lie algebra and Lb-equation.Furthermore,we get two subsidiary conclusions when we study the bimodules and dual bimodules of Leibniz algebras:one is we use the bimodule to get a necessary and sufficient condition about matched pair construction by two Leibniz algebras and the other is we use the dual bimodule to get an important nondegenerate skew-symmetric bilinear form on Leibniz algebras.
引文
[1]C.M.Bai,A unified algebraic approach to classical Yang-Baxter equation,J.Phy.A:Math.Theor.40(2007) 10073-10082.
[2]C.M.Bai,Left-symmetric bialgebras and an analogy of the classical Yang-baxter equation,Comm.Comtemp.Math.10(2008) 221-260.
[3]B.Y.Chu,Sympiectic homogeneous spaces,Trans.Amer.Math.Soc.197(1974),145-159.
[4]L.D.Faddeev,L.Takhtajan,The quantum inverse scattering method of the inverse problem and the Heisenberg XYZ model,Russ.Math.Surv.34(1979) 11-68.
[5]L.D.Faddeev,L.Takhtajan,Hamiltonian methods in the theory of solitons,Springer,Berlin(1987).
[6]Nahong Hu,Yufeng Pei,Dong Liu,A Cohomological characterization of Leibniz Central Extensions of Lie algebras,arXiv:math.QA/0605399.
[7]J.-L.Koszul,Domaines born(?)s homog(?)nes et orbites de groupes de transformations affines,Bull.Soc.Math.France 89(1961) 515-533.
[8]B.A.Kupershmidt,What a classical γ-matrix really is,J.Nonlinear Math.Phy.Vo.6(1999),448-488.
[9]M.K.Kinyou and A.Weinstein,Leibniz algebras,Courant algebroids,and multiplications on homogeneous spaces,Amer.J.Math.123(2001)525-550.
[10]M.K.Kinyon,Leibniz algebras,Lie racks and digroups,arXiV:math/0403509v5[math.RAil Aug 2006.
[11]J-L.Loday,and T.Pirashvili,The tensor categoryof linear maps and Leibniz algebras,Georgian Math.J.5(1998)263-276.
[12]J-L.Loday,Dialgebras,in "Dialgebras and related operads",Lecture Notes in Math.1763,Springer,Berlin,2001,7-66.
[13]J.M.Lodder,Leibniz cohomology for differentiable manifolds,Ann.Inst.Fourier,Grenoble 48,1(1998),73-95
[14]Y.Matsushima,Affine structures on complex mainfolds,Osaka J.Math.5(1968).215-222.(1960),731-742
[15]S.Majid,Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations,Pacific J.Math.141(1990) 311-332.
[16]J.A.Schouten,(U|¨)bet differentialkomitanten zweier kontravarianter Gr(o|¨)ssen,Nede.Akad.Were.Proc.A 43.(1940) 449-452.
[17]M.A.Semonov-Tian-Shansky,What is a classical R-matrix? Funct.Anal.Appl.17(1983) 259-272.
[18]M.Takeuchi,Matched pairs of groups and bimash products of Hopf algebras,Comm.Algebra 9(1981) 841-842.
[19]E.B.Vinberg,Convex homogeneous cones,Transl.of Moscow Math.Soc.No.12(1963) 340-403.
[2]C.M.Bai,Left-symmetric bialgebras and an analogy of the classical Yang-baxter equation,Comm.Comtemp.Math.10(2008) 221-260.
[3]B.Y.Chu,Sympiectic homogeneous spaces,Trans.Amer.Math.Soc.197(1974),145-159.
[4]L.D.Faddeev,L.Takhtajan,The quantum inverse scattering method of the inverse problem and the Heisenberg XYZ model,Russ.Math.Surv.34(1979) 11-68.
[5]L.D.Faddeev,L.Takhtajan,Hamiltonian methods in the theory of solitons,Springer,Berlin(1987).
[6]Nahong Hu,Yufeng Pei,Dong Liu,A Cohomological characterization of Leibniz Central Extensions of Lie algebras,arXiv:math.QA/0605399.
[7]J.-L.Koszul,Domaines born(?)s homog(?)nes et orbites de groupes de transformations affines,Bull.Soc.Math.France 89(1961) 515-533.
[8]B.A.Kupershmidt,What a classical γ-matrix really is,J.Nonlinear Math.Phy.Vo.6(1999),448-488.
[9]M.K.Kinyou and A.Weinstein,Leibniz algebras,Courant algebroids,and multiplications on homogeneous spaces,Amer.J.Math.123(2001)525-550.
[10]M.K.Kinyon,Leibniz algebras,Lie racks and digroups,arXiV:math/0403509v5[math.RAil Aug 2006.
[11]J-L.Loday,and T.Pirashvili,The tensor categoryof linear maps and Leibniz algebras,Georgian Math.J.5(1998)263-276.
[12]J-L.Loday,Dialgebras,in "Dialgebras and related operads",Lecture Notes in Math.1763,Springer,Berlin,2001,7-66.
[13]J.M.Lodder,Leibniz cohomology for differentiable manifolds,Ann.Inst.Fourier,Grenoble 48,1(1998),73-95
[14]Y.Matsushima,Affine structures on complex mainfolds,Osaka J.Math.5(1968).215-222.(1960),731-742
[15]S.Majid,Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations,Pacific J.Math.141(1990) 311-332.
[16]J.A.Schouten,(U|¨)bet differentialkomitanten zweier kontravarianter Gr(o|¨)ssen,Nede.Akad.Were.Proc.A 43.(1940) 449-452.
[17]M.A.Semonov-Tian-Shansky,What is a classical R-matrix? Funct.Anal.Appl.17(1983) 259-272.
[18]M.Takeuchi,Matched pairs of groups and bimash products of Hopf algebras,Comm.Algebra 9(1981) 841-842.
[19]E.B.Vinberg,Convex homogeneous cones,Transl.of Moscow Math.Soc.No.12(1963) 340-403.