L-infinity algebras and deformation theory

Subject classification: this is a mathematics resource.

A differential graded Lie algebra is a graded $k$-vector space $𝔤$ endowed with a differential $d:𝔤\to 𝔤$ of degree $+1$ and a bracket $[]:𝔤\otimes 𝔤⟶𝔤$ (of degree $0$), such that

$[]$ is skew symmetric,

$[]$ satisfies the Jacobi identity,

$d$ acts as a derivation with respect to $[]$.

We denote the cohomology of $𝔤$ with respect to $d$ by $h^{\ast }(𝔤)$. It is a graded vector space over $k$, and it inherits a Lie bracket. (In fact, it is an $L_{\mathrm{\infty }}$-algebra.)