Chen Lie Algebras

where {ΓkG}k≥1 is the lower central series (LCS) of the group, defined, inductively, by Γ1G = G and Γk+1G = (ΓkG,G), and the Lie bracket [x, y] is induced from the group commutator (x, y) = xyxy. Many properties of a group are reflected in properties of its associated graded Lie algebra. For instance, if G is finitely generated, then the abelian groups grk(G) are also finitely generated; their ranks φk(G) are important numerical invariants of G. In the case when G = Fn, the free group of rank n, Magnus showed that gr(Fn) = Ln, the free Lie algebra on n generators, whose graded ranks were computed by E. Witt. In general, though, the computation of the LCS ranks φk(G) can be exceedingly difficult.