Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures

TL;DR

This paper develops a cohomology framework for relative Rota-Baxter operators on Lie algebroids, explores their deformations, and connects them to left-symmetric algebroids and Koszul-Vinberg structures.

Contribution

It introduces a graded Lie algebra approach to characterize Rota-Baxter operators and their deformations, linking these to left-symmetric algebroids and Koszul-Vinberg structures.

Findings

Constructed a graded Lie algebra for Rota-Baxter operators on Lie algebroids.

Established a homomorphism linking Rota-Baxter and left-symmetric algebroids.

Provided cohomology theories for Koszul-Vinberg structures.

Abstract

Given a Lie algebroid with a representation, we 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 theory. 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. As applications, we give the controlling graded Lie algebra and the cohomology theory of Koszul-Vinberg structures on left-symmetric algebroids.