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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References

A. Bak, G. Donadze, N. Inassaridze and M. Ladra (2007). Homology of multiplicative Lie rings. J. Pure Appl. Algebra. 208, 761-777 P. Batten and E. Stitzinger (1996). On covers of Lie algebras. Comm. Algebra. 24, 4301-4317 B. Eick, M. Horn and S. Zandi (2012). Schur multipliers and the Lazard correspondence. Archiv der Mathematik. 99, 217-226 H. Hopf (1942). Fundamentalgruppe und zweite Bettische Gruppe. Comment. Math. Helv.. 14, 257-309 G. Karpilovsky (1987). The Schur multiplier. The Clarendon Press Oxford University Press, London Mathematical Society Monographs. New Series, New York. 2 E. I. Khukhro (1998). p-automorphisms of finite p-groups. London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge. 246 J. Schur (1904). Uber die Darstellung der endlichen Gruppen durch gebrochen lineare Substitutionen. J. Reine Angew. Math.. 127, 20-50 J. Schur (1907). Untersuchungen uber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. J. Reine Angew. Math.. 132, 85-137 C. A. Weibel (1994). Homological Algebra. Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge. 38

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History

  • Receive Date: 02 July 2013
  • Revise Date: 28 September 2013
  • Accept Date: 30 September 2013
  • Published Online: 01 June 2014

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