On the capability and Schur multiplier of nilpotent Lie algebra of class two

Niroomand, Peyman; Johari, Farangis; Parvizi, Mohsen

doi:10.1090/proc/13092

On the capability and Schur multiplier of nilpotent Lie algebra of class two
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by Peyman Niroomand, Farangis Johari and Mohsen Parvizi PDF

Proc. Amer. Math. Soc. 144 (2016), 4157-4168 Request permission

Abstract:

Recently, the authors in a joint paper obtained the structure of all capable nilpotent Lie algebras with derived subalgebra of dimension at most $1$. This paper is devoted to characterizing all capable nilpotent Lie algebras of class two with derived subalgebra of dimension $2$. It develops and generalizes the result due to Heineken for the group case.

References

Ali Reza Salemkar, Vahid Alamian, and Hamid Mohammadzadeh, Some properties of the Schur multiplier and covers of Lie algebras, Comm. Algebra 36 (2008), no. 2, 697–707. MR 2388035, DOI 10.1080/00927870701724193
Reinhold Baer, Groups with preassigned central and central quotient group, Trans. Amer. Math. Soc. 44 (1938), no. 3, 387–412. MR 1501973, DOI 10.1090/S0002-9947-1938-1501973-3
Yu. A. Bahturin, Identical relations in Lie algebras, VNU Science Press, b.v., Utrecht, 1987. Translated from the Russian by Bahturin. MR 886063
Peggy Batten, Kay Moneyhun, and Ernest Stitzinger, On characterizing nilpotent Lie algebras by their multipliers, Comm. Algebra 24 (1996), no. 14, 4319–4330. MR 1421191, DOI 10.1080/00927879608825817
Peggy Batten and Ernest Stitzinger, On covers of Lie algebras, Comm. Algebra 24 (1996), no. 14, 4301–4317. MR 1421190, DOI 10.1080/00927879608825816
Y. Berkovich, Groups of prime power order, vol. 1, de Gruyter, Berlin, 2010.
Lindsey R. Bosko, On Schur multipliers of Lie algebras and groups of maximal class, Internat. J. Algebra Comput. 20 (2010), no. 6, 807–821. MR 2726575, DOI 10.1142/S0218196710005881
F. Rudolf Beyl, Ulrich Felgner, and Peter Schmid, On groups occurring as center factor groups, J. Algebra 61 (1979), no. 1, 161–177. MR 554857, DOI 10.1016/0021-8693(79)90311-9
F. R. Beyl and J. Tappe, Extensions, Representations and the Schur Multiplicator, Lecture Notes in Mathematics 958, Berlin, Heidelberg, New York, 1989.
Serena Cicalò, Willem A. de Graaf, and Csaba Schneider, Six-dimensional nilpotent Lie algebras, Linear Algebra Appl. 436 (2012), no. 1, 163–189. MR 2859920, DOI 10.1016/j.laa.2011.06.037
G. Ellis, A non-abelian tensor product of Lie algebras. Glasg. Math. J. 39 (1991), 101-120.
Graham Ellis, Capability, homology, and central series of a pair of groups, J. Algebra 179 (1996), no. 1, 31–46. MR 1367840, DOI 10.1006/jabr.1996.0002
Willem A. de Graaf, Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2, J. Algebra 309 (2007), no. 2, 640–653. MR 2303198, DOI 10.1016/j.jalgebra.2006.08.006
Ming-Peng Gong, Classification of nilpotent Lie algebras of dimension 7 (over algebraically closed fields and R), ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–University of Waterloo (Canada). MR 2698220
Hermann Heineken, Nilpotent groups of class two that can appear as central quotient groups, Rend. Sem. Mat. Univ. Padova 84 (1990), 241–248 (1991). MR 1101296
P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130–141. MR 3389, DOI 10.1515/crll.1940.182.130
Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215
C. R. Leedham-Green and S. McKay, The structure of groups of prime power order, London Mathematical Society Monographs. New Series, vol. 27, Oxford University Press, Oxford, 2002. Oxford Science Publications. MR 1918951
Kay Moneyhun, Isoclinisms in Lie algebras, Algebras Groups Geom. 11 (1994), no. 1, 9–22. MR 1268930
Peyman Niroomand and Francesco G. Russo, A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra 39 (2011), no. 4, 1293–1297. MR 2782606, DOI 10.1080/00927871003652660
Peyman Niroomand and Francesco G. Russo, A restriction on the Schur multiplier of nilpotent Lie algebras, Electron. J. Linear Algebra 22 (2011), 1–9. MR 2770238, DOI 10.13001/1081-3810.1423
Peyman Niroomand, On dimension of the Schur multiplier of nilpotent Lie algebras, Cent. Eur. J. Math. 9 (2011), no. 1, 57–64. MR 2753882, DOI 10.2478/s11533-010-0079-3
Peyman Niroomand, Mohsen Parvizi, and Francesco G. Russo, Some criteria for detecting capable Lie algebras, J. Algebra 384 (2013), 36–44. MR 3045150, DOI 10.1016/j.jalgebra.2013.02.033
Peyman Niroomand, Characterizing finite $p$-groups by their Schur multipliers, $t(G)=5$, Math. Rep. (Bucur.) 17(67) (2015), no. 2, 249–254. MR 3375732
Peyman Niroomand, Characterizing finite $p$-groups by their Schur multipliers, C. R. Math. Acad. Sci. Paris 350 (2012), no. 19-20, 867–870 (English, with English and French summaries). MR 2990893, DOI 10.1016/j.crma.2012.10.018
Peyman Niroomand, A note on the Schur multiplier of groups of prime power order, Ric. Mat. 61 (2012), no. 2, 341–346. MR 3000665, DOI 10.1007/s11587-012-0134-4
Peyman Niroomand, The Schur multiplier of $p$-groups with large derived subgroup, Arch. Math. (Basel) 95 (2010), no. 2, 101–103. MR 2674245, DOI 10.1007/s00013-010-0154-9
Peyman Niroomand, On the order of Schur multiplier of non-abelian $p$-groups, J. Algebra 322 (2009), no. 12, 4479–4482. MR 2558872, DOI 10.1016/j.jalgebra.2009.09.030
Gregory Karpilovsky, The Schur multiplier, London Mathematical Society Monographs. New Series, vol. 2, The Clarendon Press, Oxford University Press, New York, 1987. MR 1200015
Ali Reza Salemkar, Behrouz Edalatzadeh, and Mehdi Araskhan, Some inequalities for the dimension of the $c$-nilpotent multiplier of Lie algebras, J. Algebra 322 (2009), no. 5, 1575–1585. MR 2543624, DOI 10.1016/j.jalgebra.2009.05.036
A. I. Širšov, On the bases of a free Lie algebra, Algebra i Logika Sem. 1 (1962), no. 1, 14–19 (Russian). MR 0150180
Joseph J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 538169