Cohomologies, deformations and extensions of pre-LieDer pairs

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 an arbitrary representation and using the second cohomology group to classify abelian extensions of regular pre-LieDer pairs.