Cohomologies of pre-LieDer pairs and applications

TL;DR

This paper develops a cohomology theory for pre-LieDer pairs using higher derived brackets and $L_$-algebras, and explores their deformations and extensions.

Contribution

It introduces a new cohomology framework for pre-LieDer pairs via $L_$-algebras, enabling analysis of deformations and extensions.

Findings

Defined cohomology of pre-LieDer pairs.

Characterized infinitesimal deformations by second cohomology.

Classified abelian extensions using second cohomology.

Abstract

In this paper, we use the higher derived bracket to give the controlling algebra of pre-LieDer pairs. We give the cohomology of pre-LieDer pairs by using the twist $L_\infty$-algebra of this controlling algebra. In particular, we define the cohomology of regular pre-LieDer pairs. We study infinitesimal deformations of pre-LieDer pairs, which are characterized by the second cohomology group of pre-LieDer pairs. We also define the cohomology of regular pre-LieDer pairs with coefficients in arbitrary representation and using the second cohomology group to classify abelian extensions of regular pre-LieDer pairs.