Some Studies Of Deformation Theory Of Lie-Yamaguti Algebras And Relative RB-Operators

In this thesis,we study both deformations and cohomologies of Lie-Yamaguti algebras and those of relative Rota-Baxter operators on Lie-Yamaguti algebras and linear deformations,cohomologies and Rota-Baxter-Nijenhuis structures on a 3-Lie algebra with a representation.Firstly,we introduce the notion of relative Rota-Baxter operators on Lie-Yamaguti algebras and pre-Lie-Yamaguti algebras and prove that there exists a Lie-Yamaguti algebra structure(we call it a sub-adjacent Lie-Yamaguti algebra)on a given pre-Lie-Yamaguti algebra.Moreover,a relative Rota-Baxter operator on a Lie-Yamaguti algebra gives rise to a pre-Lie-Yamaguti algebra structure on its representable space.Sequently,we introduce the notion of symplectic structure and the phase spaces on Lie-Yamaguti algebras and reveal the fact that a Lie-Yamaguti algebra owns a phase space if and only if it owns a compatible pre-Lie-Yamaguti algebra structure.Besides,we introduce the notion of Manin triples of a pre-Lie-Yamaguti algebra and prove that there is a one-to-one correspondence between it and a phase space of a Lie-Yamaguti algebra.Secondly,we study the cohomologies and deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras.We treat the cohomology of sub-adjacent LieYamaguti algebra of the relative Rota-Baxter operator as the cohomology of the relative Rota-Baxter operator.Then we characterize the deformations of relative Rota-Baxter operator by its cohomology group.In particular,linear deformations can be characterized by the 2nd-cohomology group.Thirdly,we study the linear deformations of Lie-Yamaguti algebras and prove that they are characterized by the 2nd-cohomology groups.The trivial deformations give rise to Nijenhuis operators.We introduce the notion of product(complex)structures on(real)Lie-Yamaguti algebras treating the Nijenhuis conditions as integrability conditions.We show that there exists a product(complex)structure on a(real)Lie-Yamaguti algebra if and only if it(its complxification)admits a decomposition into two subalgebras.Then we add a compatibility condition between a product structure and a complex structure to introduce the notion of a complex product structure on real Lie-Yamaguti algebras,and construct a complex product structure using pre-Lie-Yamaguti algebras.Then,we give the notions of para-Kahler structure and that of pseudo-Kahler structure on Lie-Yamaguti algebras.For para-Kahler Lie-Yamaguti algebras,we show that the decomposed two subalgebras are isotropic and introduce the notion of pseudo-Riemannian metric and its associated Levi-Civita products.We show that Levi-Civita products gives rise to a compatible pre-Lie-Yamaguti algebra structure on a para-Kahler Lie-Yamaguti algebra.For pseudo-Kahler Lie-Yamaguti algebras,we study the relationship between those and para-Kahler Lie-Yamaguti algebras.On a pseudo-Kahler Lie-Yamaguti algebra,there is a pseudo-Riemannian metric.For this pseudo-Riemannian metric,if it is positive definitely,we call this pseudo-Kahler Lie-Yamaguti algebra a real Kahler Lie-Yamaguti algebra.And we construct a real Kahler Lie-Yamaguti algebra via a pre-Lie-Yamaguti algebra.Finally,we study the linear deformation and cohomology theory of a 3-Lie algebra with a representation(we call it a 3-LieRep pair).And we prove that the linear deformations can be characterized by its second cohomology groups.Then we introduce the notion of relative Rota-Baxter-Nijenhuis structures on it.Several properties have been studied.We construct a relative Rota-Baxter-Nijenhuis structures on 3-LieRep pair from those on LieRep pairs and commutative associated algebras.