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

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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.

Related facts

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.

References

Journal references