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
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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
Additional Information
- Peyman Niroomand
- Affiliation: School of Mathematics and Computer Science, Damghan University, Damghan, Iran
- Email: niroomand@du.ac.ir, p$_$niroomand@yahoo.com
- Farangis Johari
- Affiliation: Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
- Email: farangis.johari@stu.um.ac.ir
- Mohsen Parvizi
- Affiliation: Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
- Email: parvizi@math.um.ac.ir
- Received by editor(s): August 31, 2015
- Received by editor(s) in revised form: December 21, 2015
- Published electronically: May 6, 2016
- Additional Notes: The first author acknowledges the financial support of the research council of Damghan University with the grant number 93/math/127/229.
- Communicated by: Kailash C. Misra
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4157-4168
- MSC (2010): Primary 17B30; Secondary 17B05, 17B99
- DOI: https://doi.org/10.1090/proc/13092
- MathSciNet review: 3531169