# L-infinity algebras and deformation theory

A differential graded Lie algebra is a graded ${\displaystyle k}$-vector space ${\displaystyle {\mathfrak {g}}}$ endowed with a differential ${\displaystyle d:{\mathfrak {g}}\to {\mathfrak {g}}}$ of degree ${\displaystyle +1}$ and a bracket ${\displaystyle [\,]:{\mathfrak {g}}\otimes {\mathfrak {g}}\longrightarrow {\mathfrak {g}}}$ (of degree ${\displaystyle 0}$), such that
${\displaystyle [\,]}$ is skew symmetric,
${\displaystyle [\,]}$ satisfies the Jacobi identity,
${\displaystyle d}$ acts as a derivation with respect to ${\displaystyle [\,]}$.
We denote the cohomology of ${\displaystyle {\mathfrak {g}}}$ with respect to ${\displaystyle d}$ by ${\displaystyle h^{*}({\mathfrak {g}})}$. It is a graded vector space over ${\displaystyle k}$, and it inherits a Lie bracket. (In fact, it is an ${\displaystyle L_{\infty }}$-algebra.)