### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 054, 26 pages      arXiv:2108.08906      https://doi.org/10.3842/SIGMA.2022.054

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

Meijun Liu a, Jiefeng Liu a and Yunhe Sheng b
a) School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, China
b) Department of Mathematics, Jilin University, Changchun 130012, Jilin, China

Received February 02, 2022, in final form July 07, 2022; Published online July 13, 2022

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.

Key words: cohomology; deformation; Lie algebroid; Rota-Baxter operator; Koszul-Vinberg structure; left-symmetric algebroid.

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