Second cohomology of Lie rings and the Schur multiplier

We exhibit an explicit construction for the second cohomology group H^۲(L, A) for a Lie ring L and a trivial L-module A. We show how the elements of H^۲(L, A) correspond one-to-one to the equivalence classes of central extensions of L by A, where A now is considered as an abelian Lie ring. For a finite Lie ring L we also show that H^۲(L, C^*) \cong M(L), where M(L) denotes the Schur multiplier of L. These results match precisely the analogue situation in group theory.

کلیدواژه ها:

Lie rings ، Schur multiplier of Lie rings ، central extension ، second cohomology group of Lie rings