Second cohomology of Lie rings and the Schur multiplier

Abstract

We exhibit an explicit construction for the second cohomology group$H^2(L, A)$ for a Lie ring $L$ and a trivial $L$-module $A$.We show how the elements of $H^2(L, A)$ correspond one-to-one to theequivalence classes of central extensions of $L$ by $A$, where $A$now is considered as an abelian Lie ring. For a finite Liering $L$ we also show that $H^2(L, C^*) cong M(L)$, where $M(L)$ denotes theSchur multiplier of $L$. These results match precisely the analoguesituation in group theory.

Keywords

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