L-infinity algebras and deformation theory

Subject classification: this is a mathematics resource.

A differential graded Lie algebra is a graded $k$-vector space ${\mathfrak {g}}$ endowed with a differential $d:{\mathfrak {g}}\to {\mathfrak {g}}$ of degree $+1$ and a bracket $[\,]:{\mathfrak {g}}\otimes {\mathfrak {g}}\longrightarrow {\mathfrak {g}}$ (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 ${\mathfrak {g}}$ with respect to $d$ by $h^{*}({\mathfrak {g}})$. It is a graded vector space over $k$, and it inherits a Lie bracket. (In fact, it is an $L_{\infty }$-algebra.)