The Cohomology Of A Hom-pre-Lie Algebra And Its Application

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.