Differential Graded Lie Algebras

Abstract

In this chapter we introduce the basic algebraic theory of differential graded vector spaces and differential graded Lie algebras over an arbitrary field. The application of these structures in deformation theory, for which the base field should be of characteristic 0, are postponed to Chap. 6. As in the previous chapters we assume that the reader has a basic knowledge of homological and commutative algebra, although some general and well known results are recalled in order to fix notation. Throughout this chapter any vector space and any linear map is considered over a fixed field $\mathbb {K}$. Unless otherwise specified, by the symbol $\otimes$ we mean the tensor product $\otimes _{\mathbb {K}}$ over the field $\mathbb {K}$.