The Bogomolov multiplier of Lie algebras

Year 2020, Volume: 49 Issue: 3, 1190 - 1205, 02.06.2020

Zeinab Araghi Rostani Mohsen Parvizi , Peyman Niroomand

https://doi.org/10.15672/hujms.455076

Cited By: 2

Abstract

In this paper, we extend the notion of the Bogomolov multipliers and the CP-extensions to Lie algebras. Then, we compute the Bogomolov multipliers for Abelian, Heisenberg and nilpotent Lie algebras of class at most 6. Finally, we compute the Bogomolov multipliers of complex simple and semisimple Lie algebras.

Keywords

Commutativity-Preserving exterior product, ${\tilde{B_0}}$-pairing, Curly exterior product, Bogomolov multiplier, Heisenberg Algebra

References

• [1] A. Bak, G. Donadze, N. Inassaridze and M. Ladra, Homology of multiplicatie Lie ring, J. Pure Appl. Algebra 208, 761–777, 2007.
• [2] P.G. Batten, Multipliers and covers of Lie algebras, ph. D. diss, North carolina state university, 1993.
• [3] P.G. Batten, K. Moneyhun and E. Stitzinger, On characterizing nilpotent Lie algebras by their multipliers, Comm. Algebra 24 (14), 4319-4330, 1996.
• [4] F.A. Bogomolov, The Brauer group of quotient spaces of linear representations, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 51 (3), 485–516, 688; translation in Math. USSR-Izv. 30 (3), 455-485, 1987.
• [5] E. Cartan, Sur la Reduction a sa Forme Canonique de la Structure d’un Groupe de Transformations Fini et Continu, (French) Amer. J. Math. 18 (1) 1–61, 1896.
• [6] Y. Chen and R. Ma, Some groups of order p6 with trivial Bogomolov multipliers, arxiv: 1302. 0584v5.
• [7] S. Cicalo, W.A. de Graaf and C. Schneider, Six-dimensional nilpotent Lie algebras, Linear Algebra Appl. 436 (1), 163–189, 2012.
• [8] G. Ellis, Nonabelian exterior products of Lie algebras and an exact sequence in the homology of Lie algebras, J. Pure Appl. Algebra 46 (2-3), 111-115, 1987.
• [9] W.A. de Graaf, Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2, J. Algebra 309 (2), 640–653, 2007.
• [10] B.C. Hall, Lie groups, Lie algebras and representations, An elementary introduction. Graduate Texts in Mathematics, 222, Springer-Verlag, New York, xiv+351 pp, 2003.
• [11] P. Hardy, On characterizing nilpotent Lie algebras by their multipliers. III, Comm. Algebra 33 (11), 4205-4210, 2005.
• [12] P. Hardy and E. Stitzinger, On characterizing nilpotent Lie algebras by their multipliers t(L) = 3, 4, 5, 6, Comm. Algebra 26 (11), 3527-3539, 1998.
• [13] U. Jezernik and P. Moravec, Universal commutator relations, Bogomolov multipliers and commuting probability, J. Algebra 428, 1–25, 2015.
• [14] U. Jezernik and P. Moravec, Commutativity preserving extensions of groups, Proc. Roy. Soc. Edinburgh Sect. A 148 (3), 575–592, 2018.
• [15] M.R. Jones, Multiplicators of p-groups, Math. Z. 127, 165-166, 1972.
• [16] M. Kang, Bogomolov multipliers and retract rationality for semidirect products, J. Algebra 397, 407–425, 2014.
• [17] A.W. Knapp, Lie groups beyond an introduction, Second edition. Progress in Mathematics, 140. Birkhäuser Boston, Inc., Boston, MA, xviii+812 pp, 2002.
• [18] B. Kunyavskii, The Bogomolov multiplier of finite simple groups, Cohomological and geometric approaches to rationality problems, 209–217, Progr. Math, 282, Birkhauser Boston, Inc, Boston, MA, 2010.
• [19] M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, (French) Ann. Sci. Ecole Norm. Sup. 71 (3), 101-190, 1954.
• [20] S. Lie, Theorie der Transformationsgruppen I, (German) Math. Ann. 16 (4), 441-528, 1880.
• [21] K. Moneyhun, Isoclinisms in Lie algebras, Algebras Groups Geom, (English summary) Algebras Groups Geom. 11 (1), 9-22, 1994.
• [22] P. Moravec, Unramified Brauer groups of finite and infinite groups, Amer. J. Math. 134 (6), 1679–1704, 2012.
• [23] P. Niroomand, On dimension of the Schur multiplier of nilpotent Lie algebras, Cent. Eur. J. Math. 9 (1), 57–64, 2011.
• [24] P. Niroomand and M. Parvizi, 2-Nilpotent multipliers of a direct product of Lie algebras, Rend. Circ. Mat. Palermo (2) 65 (3), 519-523, 2016.
• [25] D.J. Saltman, Noether’s problem over an algebraically closed field, Invent. Math. 77 (1), 71–84, 1984.
• [26] H. Samelson, Notes on Lie algebras, Van Nostrand Reinhold Mathematical Studies, No. 23 Van Nostrand Reinhold Co., New York- London-Melbourne, vi+165 pp, (loose errata), 1969.
• [27] V.S. Varadarajan, Lie groups, Lie algebras and their representations, Reprint of the 1974 edition. Graduate Texts in Mathematics, 102, Springer-Verlag, New York, 1984.
• [28] J.B. Zuber, Invariances in physics and group theory, Sophus Lie and Felix Klein: the Erlangen program and its impact in mathematics and physics, 307-326, IRMA Lect. Math. Theor. Phys. 23, Eur. Math. Soc. Zürich, 2015.