Second cohomology of Lie rings and the Schur multiplier

• Main Authors: Max Horn, Seiran Zandi
• Format: Article
• Language: English
• Published: University of Isfahan 2014-06-01
• Series: International Journal of Group Theory
• Subjects: Lie rings; Schur multiplier of Lie rings; central extension; second cohomology group of Lie rings
• Online Access: http://www.theoryofgroups.ir/?_action=showPDF&article=3589&_ob=cf135f0fb1340cca48124481e4a34726&fileName=full_text.pdf.

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.