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

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.