The lower central series of the free partially commutative group (Q1204126)

Let \(\theta\) be a subset of \(A \times A\). The free partially commutative group \(F(A,\theta)\) is the quotient of the free group generated by \(A\) by the relations \(ab = ba\), \((a,b) \in \theta\). The Magnus transformation \(\mu : F(A,\theta) \to \mathbb{Z} \langle\langle A,\theta \rangle\rangle\) is the group homomorphism which sends \(a\) onto \(1 + a\), where \(\mathbb{Z} \langle\langle A,\theta \rangle\rangle\) is the \(\mathbb{Z}\)-algebra of series in the variables \(a \in A\), subject to the same relations. Let \((F_ n(A,\theta))\) be the lower central series of \(F(A,\theta)\). The direct sum \(\bigoplus_{n \in \mathbb{N}} F_ n(A,\theta)/F_{n + 1}(A,\theta)\) has a natural structure of a Lie algebra, and is isomorphic to the free partially commutative Lie algebra \(L(A,\theta)\). An element \(g\) of \(F(A,\theta)\) is in \(F_ n\) if and only if \(\mu(g) = 1 + O(A^ n)\); \(F(A,\theta)\) has an order compatible with its group structure.