Differential graded Lie algebra

In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible. Such objects have applications in deformation theory and rational homotopy theory.
Definition.
A differential graded Lie algebra is a graded vector space formula_1 over a field of characteristic zero together with a bilinear map formula_2 and a differential formula_3 satisfying
for any homogeneous elements "x", "y" and "z" in "L". Notice here that the differential lowers the degree and so this differential graded Lie algebra is considered to be homologically graded. If instead the differential raised degree the differential graded Lie algebra is said to be cohomologically graded (usually to reinforce this point the grading is written in superscript: formula_5). The choice of cohomological grading usually depends upon personal preference or the situation as they are equivalent: a homologically graded space can be made into a cohomological one via setting formula_6.
Alternative equivalent definitions of a differential graded Lie algebra include:
A morphism of differential graded Lie algebras is a graded linear map formula_8 that commutes with the bracket and the differential, i.e., formula_9 and formula_10. Differential graded Lie algebras and their morphisms define a category.
Products and coproducts.
The product of two differential graded Lie algebras, formula_11, is defined as follows: take the direct sum of the two graded vector spaces formula_12, now equip it with the bracket formula_13 and differential formula_14.
The coproduct of two differential graded Lie algebras, formula_15, is often called the free product. It is defined as the free graded Lie algebra on the two underlying vector spaces with the unique differential extending the two original ones.
Connection to deformation theory.
The main application is to the deformation theory over fields of characteristic zero (in particular over the complex numbers.) The idea goes back to Daniel Quillen's work on rational homotopy theory. One way to formulate this thesis (due to Vladimir Drinfeld, Boris Feigin, Pierre Deligne, Maxim Kontsevich, and others) might be:
A Maurer-Cartan element is a degree formula_16 element, formula_17, that is a solution to the Maurer–Cartan equation: