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Horn, Max, Zandi, Seiran. (1393). Second cohomology of Lie rings and the Schur multiplier. , 3(2), 9-20. doi: 10.22108/ijgt.2014.3589
Max Horn; Seiran Zandi. "Second cohomology of Lie rings and the Schur multiplier". , 3, 2, 1393, 9-20. doi: 10.22108/ijgt.2014.3589
Horn, Max, Zandi, Seiran. (1393). 'Second cohomology of Lie rings and the Schur multiplier', , 3(2), pp. 9-20. doi: 10.22108/ijgt.2014.3589
Horn, Max, Zandi, Seiran. Second cohomology of Lie rings and the Schur multiplier. , 1393; 3(2): 9-20. doi: 10.22108/ijgt.2014.3589

Second cohomology of Lie rings and the Schur multiplier

International Journal of Group Theory
مقاله 2، دوره 3، شماره 2، شهریور 2014، صفحه 9-20    اصل مقاله (311.25 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2014.3589
نویسندگان
Max Horn1؛ Seiran Zandi* 2
1AG Algebra, Mathematisches Institut, Justus-Liebig-Universität Gießen, Arndtstrasse 2 35392, Giessen, Germany
2Department of Mathematics, University of Kharazmi, P.O.Box 15614, Tehran, Iran
چکیده
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.
کلیدواژه‌ها
Lie rings؛ Schur multiplier of Lie rings؛ central extension؛ second cohomology group of Lie rings
مراجع
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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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