A differential graded Lie algebra is a graded -vector space
endowed with a differential of degree and a bracket
(of degree ), such that
- is skew symmetric,
- satisfies the Jacobi identity,
- acts as a derivation with respect to .
We denote the cohomology of with respect to by . It is a graded vector space over , and it inherits a Lie bracket. (In fact, it is an -algebra.)