# Associated graded Lie algebra of braid groups

Let $G$ be a group and let $\Gamma_G(k)$ be the $k$th term of the lower central series of $G$. For each $k\geq 1$, set $\mathcal{L}_G(k)=\Gamma_G(k)/\Gamma_G(k+1)$ and $$\mathcal{L}_G:=\bigoplus_{k\geq 1}\mathcal{L}_G(k).$$ Then $\mathcal{L}_G$ has a graded Lie algebra structure induced from the commutator bracket on $G$.

I am looking for the known results about $\mathcal{L}_G$ where $G$ is a braid group or reduced braid group (the factor group of braid group such that each string is allowed to intersect itself). I think that F.R. Cohen has some work on this, but I am not sure in which paper. Any references will be much appreciated.

Take a look at the following papers

F.R. Cohen - S. Prassidis: "On injective homomorphisms for pure braid groups, and associated Lie algebras", J. Algebra 298 (2006), no. 2, 363–370.

and

F.R. Cohen - J. Wu: "On braid groups and homotopy groups", Groups, homotopy and configuration spaces, 169–193, Geom. Topol. Monogr., 13, Geom. Topol. Publ., Coventry, 2008.