Second cohomology of Lie rings and the Schur multiplier

Document Type: Research Paper

Authors

1 AG Algebra, Mathematisches Institut, Justus-Liebig-Universität Gießen, Arndtstrasse 2 35392, Giessen, Germany

2 Department of Mathematics, University of Kharazmi, P.O.Box 15614, Tehran, Iran

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 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 $H^2(L, C^*) \cong M(L)$, where $M(L)$ denotes the Schur multiplier of $L$. These results match precisely the analogue situation in group theory.

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