An Introduction to Pre‐Lie Algebras

Abstract

This chapter gives a brief introduction to pre-Lie algebras, with emphasis on their connections with some related structures. Pre-Lie algebras have several other names: left-symmetric algebra, right-symmetric algebra, pre-Lie algebra, Quasi-associative algebra, and Vinberg algebra or Koszul algebra or Koszul–Vinberg algebra. Some subclasses of pre-Lie algebras are very interesting. Some of them were even introduced and then developed independently. The chapter introduces some background information and different motivations of introducing the notion of pre-Lie algebra. It also introduces some basic properties of pre-Lie algebras including some comments on the studies of structure theory and representation theory and some classification results. The chapter gives the constructions of pre-Lie algebras from some known structures such as commutative associative algebras, Lie algebras, associative algebras and linear functions. It focuses on the close relationship between pre-Lie algebras and the classical Yang–Baxter equation.