【作者】 刘珊珊；

【导师】 生云鹤；

【作者基本信息】 吉林大学 ， 基础数学， 2021， 博士

【摘要】 在这篇论文中,我们研究了 Hom-预李代数及其上同调理论.作为Hom-预李代数的上同调理论的应用,我们研究了 Hom-预李代数的形变理论,扩张理论和Hom-预李双代数.首先,我们定义了 Hom-预李代数的表示,并且给出了 Hom-预李代数的上同调.我们定义了 Hom-预李代数的线性形变,它是由Hom-预李代数的相对于正则表示的二阶同调群所刻画.我们定义了 Hom-预李代数的Nijenhuis算子,并且证明了一个Hom-预李代数的平凡的线性形变给出了一个Hom-预李代数的Nijenhuis算子,反过来,一个Hom-预李代数的Nijenhuis算子产生一个Hom-预李代数的平凡的线性形变.我们定义了 Hom-预李代数的O-算子和Hessian结构,一个Hom-预李代数的相对于正则表示的对偶表示的O-算子可以给出一个Hom-预李代数的Hessian结构,反过来,一个Hom-预李代数的Hessian结构给出一个Hom-预李代数的相对于正则表示的对偶表示的O-算子.其次,我们定义了 Hom-预李代数的Manin triple和Hom-预李双代数.我们证明了Hom-预李代数的相容对,Manin triple和Hom-预李双代数三者相互等价.根据Hom-预李代数的上同调理论,我们定义了上边缘Hom-预李双代数和Hom-s-矩阵,并且证明了一个Hom-s-矩阵能够自然的产生一个上边缘Hom-预李双代数.我们研究了 Hom-预李代数的Hom-O-算子,并且证明关于对偶表示的半直积Hom-预李代数的Hom-s-矩阵给出一个Hom-预李代数的Hom-O-算子,反过来,一个Hom-预李代数的Hom-O-算子给出一个关于对偶表示的半直积Hom-预李代数的Hom-s-矩阵.我们定义了 Hom-预李代数背后的代数结构Hom-L-dendriform代数,一个Hom-预李代数的可逆的Hom-O-算子给出一个兼容的Hom-L-dendriform代数,反之,一个兼容的Hom-L-dendriform代数自然的给出一个Hom-预李代数的Hom-O-算子.再次,为了研究Hom-预李代数的乘积和代数同态的同时形变,我们引入了Hom-预李代数的完全上同调的概念.我们定义了Hom-预李代数的同时形变,Hom-预李代数的同时形变可以由Hom-预李代数的相对于正则表示的完全上同调的二阶同调群所刻画.我们研究了 Hom-预李代数的交换扩张,并且证明了 Hom-预李代数的交换扩张被Hom-预李代数的完全上同调的二阶同调群分类.最后,我们研究了Hom-泊松代数和Hom-预泊松代数,并且证明了由Hom-zinbiel代数的Hom-dendriform形式形变可以得到一个Hom-预泊松代数.该Hom-预泊松代数称为形式形变后的Hom-deudriform代数的半经典极限,该Hom-dendriform代数称为Hom-zinbiel代数的形变量子化.我们定义了 Hom-预泊松代数的Hom-O-算子,并且证明了一个Hom-泊松代数的可逆的Hom-O-算子给出一个兼容的Hom-预泊松代数.反之,一个Hom-预泊松代数自然的给出了它的邻接Hom-泊松代数的Hom-O-算子.我们定义了 Hom-pre-Gerstenhaber代数,证明 了一个Hom-pre-Gerstenhaber代数可以给出一个Hom-Gerst.enhaber代数.我们研究了 Hom-Aguiar预泊松代数和Hom-泊松代数的Hom-平均算子,证明了一个正则Hom-泊松代数的Hom-平均算子给出一个Hom-Aguiar预泊松代数.更多还原

【Abstract】 In this paper,we study Hom-pre-Lie algebras and the cohomology theory of Hom-pre-Lie algebras.As applications,we study deformations,extensions and Hom-pre-Lie bialgebrasFirst,we define the representations of Hom-pre-Lie algebras,cohomology theory of Hom-pre-Lie algebras.We define linear deformations of Hom-pre-Lie algebras,which are characterized by the second cohomology groups of Hom-pre-Lie algebras with the coefficients in the regular representations.We define a Nijenhuis operator on a,Hom-pre-Lie algebra and prove a trivial linear deformation of a Hom-pre-Lie algebra can generate a Nijenhuis operator on a Hom-pre-Lie algebra.On the contrary,a Nijenhuis operator on a Hom-pre-Lie algebra can generate a trivial linear deformation of a Hom-pre-Lie algebra.We introduce the notion of an O-operator and a Hessian structure on a Hom-pre-Lie algebra.A O-operator of a Hom-pre-Lie algebra with the coefficients in the dual representations of regular representations can generate a Hessian structure on a Hom-pre-Lie algebra.On the contrary,a Hessian structure on a Hom-pre-Lie algebra can generate a O-operator of a Hom-pre-Lie algebra with the coefficients in the dual representations of regular representations.Second,We introduce the notion of Manin triples for Hom-pre-Lie algebras and Hom-pre-Lie bialgebras.We show that certain matched pairs of Hom-pre-Lie algebras,standard Manin triples for Hom-pre-Lie algebras and Hom-pre-Lie bialgebras are equivalent.Due to the usage of the cohomology theory,it makes us successfully study coboundary Hom-pre-Lie bialgebras and Hom-s-matrix.We show that a Hom-s-matrix can construct coboundary Hom-pre-Lie bialgebras naturally.We introduce the notions of Hom-O-operators on Hom-pre-Lie algebras.We prove a Hom-s-matrix of a semi-product Hom-pre-Lie algebra of a dual representation generate a Hom-O-operator on a Hom-pre-Lie algebra.On the contrary,a Hom-O-operator on a Hom-pre-Lie algebra generate a Hom-s-matrix of a semi-product Hom-pre-Lie algebra of a dual representation.We define Hom-L-dendriform algebras.There exists a compatible Hom-L-dendriform algebra structure if and only if there exists an invertible Hom-O-operator on Hom-pre-Lie algebras.Third,in order to study the simultaneous deformations of the products and homo-morphisms of Hom-pre-Lie algebras,we also introduce the notion of the full cohomology of Hom-pre-Lie algebras.We define the simultaneous deformations of Hom-pre-Lie alge-bras.The simultaneous deformations of Hom-pre-Lie algebras are characterized by the second cohomology groups of Hom-pre-Lie algebras with the coefficients in the regular representations.We also study the abelian extensions of Hom-pre-Lie algebras,which are characterized by the second cohomology groups of Hom-pre-Lie algebras.Finally,We study Hom-Poisson algebras and Hom-pre-Poisson algebras.By a Hom-dendriform formal deformations of a Hom-zinbiel algebra,we get a Horu-pre-Poisson algebra.,which is called the semi-classical limit of the Hom-dendriform algebra,and the Hom-dendriform algebra is called the Hom-dendriform deformation quantization of the Hom-zinbiel algebra.We define a Hom-O-operator on a Hom-Poisson algebra,which gives a compatible Hom-pre-Poisson algebra.On the contrary,a Hom-pre-Poisson algebra gives a Hom-O-operator on its sub-adjacent Hom-Poisson algebra naturally.We study Hom-pre-Gerstenhaber algebras and prove a Hom-pre-Gerstenhaber algebras give rise to a Hom-Gerstenhaber algebras.We study Hom-Aguiar-pre-Poisson algebras and Hom-average-operators on Hom-Poisson algebras and we prove a Hom-average-operator on a Hom-Poisson algebra gives a Hom-Aguiar-pre-Poisson algebra.更多还原