The Cohomology Theory Of Relative Rota-Baxter Operators On Lie Algebroids And Pre-Lie-Rinehart Algebras

Lie algebroid is an infinite dimensional Lie algebra,which is a generalization of the Lie algebras and tangent bundles.Lie-Rinehart algebra is an algebraic generalization of the Lie algebroid.They are presented as a description of objects in algebras and geometry,such as foli-ation structures and Lie algebra actions.And they have important applications in mathematical physics such as gauge theory,string theory etc.This thesis mainly studies the cohomology theories of the relative Rota-Baxter operators on Lie algebroids and pre-Lie-Rinehart algebras,as well as their applications in deformations and extensions.It consists of six chapters.In Chapter 1,we introduce the backgrounds and recent developments of the research work in this thesis,then describe the main results.In Chapter 2,we review representations and cohomology theories of Lie algebroids,and introduce graded Lie algebras,L_∞-algebras and their Maurer-Cartan elements.In Chapter 3,we study deformations and cohomologies of relative Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures.Given a Lie algebroid with a representation,we firstly construct a graded Lie algebra whose Maurer-Cartan elements characterize relative Rota-Baxter operators on Lie algebroids.We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extendability of order n deformations to order(n+1)deformations of relative Rota-Baxter operators in terms of this cohomology the-ory.Secondly,we also construct a graded Lie algebra on the space of multi-derivations of a vector bundle whose Maurer-Cartan elements characterize left-symmetric algebroids.We show that there is a homomorphism from the controlling graded Lie algebra of relative Rota-Baxter operators on Lie algebroids to the controlling graded Lie algebra of left-symmetric algebroids.Consequently,there is a natural homomorphism from the cohomology groups of a relative Rota-Baxter operator to the deformation cohomology groups of the associated left-symmetric algebroid.Finally,we give the controlling graded Lie algebras and the cohomology theory of Koszul-Vinberg structures on left-symmetric algebroids as applications.In Chapter 4,we study cohomologies and deformations of relative Rota-Baxter operators of nonzero weight on Lie algebroids.We introduce the notions of relative Rota-Baxter operators of nonzero weight on Lie algebroids.Then we construct a differential graded Lie algebra whose Maurer-Cartan elements are relative Rota-Baxter operators of nonzero weight.We give a cohomology theory for a relative Rota-Baxter operator of nonzero weight on Lie algebroids and use the first cohomology group to classify infinitesimal deformations.We show that a relative Rota-Baxter operator of nonzero weight on Lie algebroid induces a post-Lie algebroid naturally.We study cohomologies of post-Lie algebroids and show that abelian extensions of post-Lie algebroids are classified by the second cohomology groups.In Chapter 5,we study cohomologies and deformations of twisted relative Rota-Baxter operators on Lie algebroids.First,we introduce the notions of twisted relative Rota-Baxter operators of on Lie algebroids.We construct a special L_∞-algebra by using the representations of Lie algebroids and higher-order derived brackets.We show that Maurer-Cartan elements of this special L_∞-algebra are twisted relative Rota-Baxter operators on Lie algebroids.Then we give a cohomology theory for twisted relative Rota-Baxter operators on Lie algebroids and use the first cohomology group to classify infinitesimal deformations.Finally,We show that a twisted relative Rota-Baxter operator naturally induces a NS-Lie algebroid.In Chapter 6,we study cohomologies and crossed modules for pre-Lie Rinehart alge-bras.We construct pre-Lie-Rinehart algebras from r-matrices through Lie algebra actions.We study cohomologies of pre-Lie-Rinehart algebras and show that abelian extensions of pre-Lie-Rinehart algebras are classified by the second cohomology groups.We introduce the notion of crossed modules for pre-Lie-Rinehart algebras and show that they are classified by the third cohomology groups of pre-Lie-Rinehart algebras.At last,we use(pre-)Lie-Rinehart 2-algebras to characterize the crossed modules for(pre-)Lie Rinehart algebras.