Nilpotent multiplier of abelian group is graded component of free Lie ring

Statement

Suppose $G$ is an abelian group and $c$ is a positive integer. Denote by $\mathcal{L}(G)$ the free Lie ring on $G$. Note that $\mathcal{L}(G)$ naturally has the structure of a graded Lie ring. The claim is that the $c$-nilpotent multiplier $M^{(c)}(G)$ (defined as the Baer invariant of $G$ with respect to the subvariety of the variety of groups given by groups of nilpotency class at most $c$) is isomorphic, as an abelian group, to the $(c+1)^{th}$ graded component of $\mathcal{L}(G)$:

$M^{(c)}(G) \cong (c+1)^{th} \mbox{ graded component of } \mathcal{L}(G)$

Note that the first graded component is $G$, so we get that:

$\mbox{Additive group of } \mathcal{L}(G) \cong G \oplus \bigoplus_{c=1}^\infty M^{(c)}(G)$

Interpretation in Schur functor terms

The graded components of the free Lie ring of $G$ can be viewed as functors of $G$. Each of these functors can explicitly described in Schur functor terms: see induction formula for Lie operad.

Particular cases

Value of $c$ Term for group of class at most $c$ Term for $c$-nilpotent multiplier Statement How it follows
1 abelian group Schur multiplier $M(G)$ Schur multiplier of abelian group is its exterior square We have $M(G) = M^{(1)}(G)$ is the degree two graded component of $\mathcal{L}(G)$. This can be described as the quotient of $G \otimes G$ by all the degree two relations. The degree two relations are all generated by alternation, so the quotient is the exterior square $G \wedge G$.
2 group of nilpotency class two 2-nilpotent multiplier $M^{(2)}(G)$ Explicit description of 2-nilpotent multiplier of abelian group in terms of relations of the tensor cube $M^{(2)}(G)$ is the degree three graded component of $\mathcal{L}(G)$. This is a quotient of $G \otimes G \otimes G$ by a bunch of relations: alternation in the inner two variables and Jacobi identity in all three variables. The quotient by these relations is the 2-nilpotent multiplier.