Schur multiplier of a Lie ring
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.
|1||second homology group||It is the second homology group for trivial Lie ring action .|
|2||second cohomology group||(?) It is the second cohomology group for trivial Lie ring action of on , i.e., the group . Note that we could replace by any divisible abelian group viewed as an abelian Lie ring.|
|3||Baer invariant||It is the Baer invariant of 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 given by . Here is the exterior square of and is the derived subring of .|
|5||Hopf's formula||It is given by Hopf's formula for Schur multiplier: If is isomorphic to the quotient of a free Lie ring by an ideal , then .|