Given a Lie algebra g, with $Ug$ being its universal enveloping algebra, one can construct a cochain complex $d: Ug^n \rightarrow Ug^{n+1}$, and a Gerstenhaber bracket on $\oplus_n Ug^n$ so that $\oplus_n Ug^n$ becomes a dgla.

Is there an analogus construction for a $L_\infty$ algebra $g$? When g is a dgla, it seems a similar construction works. Does anyone know if there is an explicit construction for a $L_3$-algebra (all higher brackets vanish except for $l_1$, $l_2$ and $l_3$)?

Many thanks.