Deformations and abelian extensions of compatible pre-Lie algebras

TL;DR

This paper explores the structure, deformations, and extensions of compatible pre-Lie algebras, introducing new cohomological tools and establishing their relation to compatible Lie algebras.

Contribution

It develops a cohomology theory for compatible pre-Lie algebras, constructs a controlling graded Lie algebra, and classifies abelian extensions via second cohomology.

Findings

Maurer-Cartan elements characterize compatible pre-Lie structures

Second cohomology classifies abelian extensions

Trivial second cohomology implies rigidity

Abstract

In this paper, we first give the notation of a compatible pre-Lie algebra and its representation. We study the relation between compatible Lie algebras and compatible pre-Lie algebras. We also construct a new bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible pre-Lie structures. We give the bidifferential graded Lie algebra which controls deformations of a compatible pre-Lie algebra. Then, we introduce a cohomology of a compatible pre-Lie algebra with coefficients in itself. We study infinitesimal deformations of compatible pre-Lie algebras and show that equivalent infinitesimal deformations are in the same second cohomology group. We further give the notion of a Nijenhuis operator on a compatible pre-Lie algebra. We study formal deformations of compatible pre-Lie algebras. If the second cohomology group $\huaH^2(\g;\g)$ is trivial, then the compatible pre-Lie algebra is rigid. Finally, we give a cohomology of a compatible pre-Lie algebra with coefficients in arbitrary representation and study abelian extensions of compatible pre-Lie algebras using this cohomology. We show that abelian extensions are classified by the second cohomology group.