Schur multiplier of a Lie ring

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Definition

The notion of Schur multiplier of a Lie ring is defined analogously to the notion of Schur multiplier of a group. In fact, each of the equivalent formulations of the definition of Schur multiplier of a group has an analogous definition in the Lie ring context.

No. Shorthand Full definition
1 second homology group It is the second homology group for trivial Lie ring action \! H_2(L;\mathbb{Z}).
2 second cohomology group (?) It is the second cohomology group for trivial Lie ring action of L on \mathbb{C}^*, i.e., the group \! H^2(L,\mathbb{C}^*). Note that we could replace \mathbb{C}^* by any divisible abelian group viewed as an abelian Lie ring.
3 Baer invariant It is the Baer invariant of G corresponding to the subvariety of abelian Lie rings in the variety of Lie rings.
4 Kernel of Lie bracket It is the kernel of the Lie bracket homomorphism from exterior square to derived subgroup, i.e., it is the kernel of the map L \wedge L \to [L,L] given by x \wedge y \mapsto [x,y]. Here L \wedge L is the exterior square of L and [L,L] is the derived subring of L.
5 Hopf's formula It is given by Hopf's formula for Schur multiplier: If L is isomorphic to the quotient of a free Lie ring F by an ideal R, then M(G) = (R \cap [F,F])/[R,F].