分裂的正则双Hom-Poisson代数结构
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摘要
作为分裂的正则Hom-Poisson代数的自然推广,介绍了一类分裂的正则双Hom-Poisson代数.利用这类代数根连通的发展技巧,证明了分裂的正则双Hom-Poisson代数B可写成■,其中U为极大α交换子代数H的子空间,I_([α])为B的理想,若[α]≠[β],则满足[I_([α]), I_([β])]+I_([α])I_([β])=0.在一定条件下,描述了B的最大长度和它的半单性.
We introduce the class of split regular biHom-Poisson algebras as the natural generalization of split regular Hom-Poisson algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular biHom-Poisson algebras B is of the form B = U +∑_αI_([α]) with U a subspace of a maximal abelian subalgebra H and any I_([α]), a well described ideal of B, satisfying [I_([α]), I_([β])] + I[α]I_([β]) = 0 if [α]≠[β]. Under certain conditions, in the case of B being of maximal length, the simplicity of the algebra is characterized.
We introduce the class of split regular biHom-Poisson algebras as the natural generalization of split regular Hom-Poisson algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular biHom-Poisson algebras B is of the form B = U +∑_αI_([α]) with U a subspace of a maximal abelian subalgebra H and any I_([α]), a well described ideal of B, satisfying [I_([α]), I_([β])] + I[α]I_([β]) = 0 if [α]≠[β]. Under certain conditions, in the case of B being of maximal length, the simplicity of the algebra is characterized.
引文
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[6]Graziani G,Makhlouf A,Menini C,et al.BiHom-associative algebras,BiHom-Lie algebras and BiHom-bialgebras[J].Symmetry Integrability and Geometry:Methods and Applications,2015,11(1):1-34.
[7]Calderón A J.On split Lie algebras with symmetric root systems[J].Proceedings of the Indian Academy of Sciences:Mathematical Sciences,2008,118(3):351-356.DOI:10.1007/s12044-008-0027-3.
[8]Aragon M J,Calderón A J.Split regular Hom-Lie algebras[J].Journal of Lie Theory,2015,25(3):875-888.
[9]Calderón A J,Sánchez J M.The structure of split regular BiHom-Lie algebras[J].Journal of Geometry and Physics,2016,110:296-305.DOI:10.1016/j.geomphys.2016.08.004.
[10]Aragon M J,Calderón A J.The structure of split regular Hom-Poisson algebras[J].Colloquium Mathematicum,2016,145(1):1-13.
[2]刘翠梅,李彦敏,傅景礼.非保守动力系统的Lie对称性代数[J].云南大学学报:自然科学版,2008,30(4):371-375.Liu C M,Li Y M,Fu J L.Lie symmetry algebraic of nonconservat ive dynamical systems[J].Journal of Yunnan University:Natural Sciences Editions,2008,30(4):371-375.
[3]Makhlouf A,Silvestrov S D.Hom-algebra structures[J].Journal of Generalized Lie Theory and Applications,2008,2(2):51-64.DOI:10.4303/jglta/S070206.
[4]Makhlouf A,Silvestrov S D.Hom-algebras and Hom-coalgebras[J].Journal of Algebra and Its Applications,2010,9(4):553-589.DOI:10.1142/S0219498810004117.
[5]Makhlouf A,Silvestrov S D.Notes on formal deformations of Hom-associative and Hom-Lie algebras[J].Forum Mathematicum,2010,22(4):715-739.
[6]Graziani G,Makhlouf A,Menini C,et al.BiHom-associative algebras,BiHom-Lie algebras and BiHom-bialgebras[J].Symmetry Integrability and Geometry:Methods and Applications,2015,11(1):1-34.
[7]Calderón A J.On split Lie algebras with symmetric root systems[J].Proceedings of the Indian Academy of Sciences:Mathematical Sciences,2008,118(3):351-356.DOI:10.1007/s12044-008-0027-3.
[8]Aragon M J,Calderón A J.Split regular Hom-Lie algebras[J].Journal of Lie Theory,2015,25(3):875-888.
[9]Calderón A J,Sánchez J M.The structure of split regular BiHom-Lie algebras[J].Journal of Geometry and Physics,2016,110:296-305.DOI:10.1016/j.geomphys.2016.08.004.
[10]Aragon M J,Calderón A J.The structure of split regular Hom-Poisson algebras[J].Colloquium Mathematicum,2016,145(1):1-13.