On the CP exterior product of Lie algebras

Document Type : Original Article


Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran


In this paper, under certain conditions, we show that the non-abelian CP exterior product distributes over direct product of Lie algebras. Then we present some properties about CP extension of Lie algebras.


Main Subjects

[1] Z. Araghi Rostami, M. Parvizi, and P. Niroomand, Bogomolov multiplier and the lazard correspondence, Commun. in Alg., 48 (2020), pp. 1201–1211.
[2] , The bogomolov multiplier of lie algebras, Hacet. J. Math. Stat., 49 (2020), pp. 1190–1205.
[3] Y. G. Berkovich, On the order of the commutator subgroup and the schur multiplier of a finite p-group, J. Alg., 144 (1991), pp. 269–272.
[4] F. A. Bogomolov, The brauer group of quotient spaces by linear group actions, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 51, 688, no. 3, 1987, 485-516, Mathematics of the USSR-Izvestiya, 30 (1988), p. 455.
[5] E. Cartan, Sur la Reduction a sa Forme Canonique de la Structure d’un Groupe de Transformations Fini et Continu, Amer. J. Math., 18 (1896), pp. 1–61.
[6] Y. Chen and R. Ma, Bogomolov multipliers of some groups of order p 6 , Comm. Algebra, 49 (2021), pp. 242– 255.
[7] S. Cicalo, W. A. de Graaf, and C. Schneider ` , Six-dimensional nilpotent Lie algebras, Linear Algebra Appl., 436 (2012), pp. 163–189.
[8] G. J. Ellis, Nonabelian exterior products of Lie algebras and an exact sequence in the homology of Lie algebras, J. Pure Appl. Algebra, 46 (1987), pp. 111–115.
[9] P. Hardy and E. Stitzinger, On characterizing nilpotent Lie algebras by their multipliers, t(L) = 3, 4, 5, 6, Comm. Algebra, 26 (1998), pp. 3527–3539.
[10] U. Jezernik and P. Moravec, Universal commutator relations, Bogomolov multipliers, and commuting probability, J. Algebra, 428 (2015), pp. 1–25.
[11] , Commutativity preserving extensions of groups, Proc. Roy. Soc. Edinburgh Sect. A, 148 (2018), pp. 575– 592.
[12] M. R. Jones, Multiplicators of p-groups, Math. Z., 127 (1972), pp. 165–166.
[13] M.-c. Kang, Bogomolov multipliers and retract rationality for semidirect products, J. Algebra, 397 (2014), pp. 407–425.
 [14] A. W. Knapp, Lie groups beyond an introduction, vol. 140 of Progress in Mathematics, Birkh¨auser Boston, Inc., Boston, MA, second ed., 2002.
[15] B. Kunyavski˘ı, The Bogomolov multiplier of finite simple groups, in Cohomological and geometric approaches to rationality problems, vol. 282 of Progr. Math., Birkh¨auser Boston, Boston, MA, 2010, pp. 209–217.
[16] M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. Ecole Norm. Sup. (3), 71 (1954), pp. 101–190.
[17] S. Lie, Theorie der Transformationsgruppen I, Math. Ann., 16 (1880), pp. 441–528.
[18] K. Moneyhun, Isoclinisms in Lie algebras, Algebras Groups Geom., 11 (1994), pp. 9–22.
[19] P. Moravec, Unramified Brauer groups of finite and infinite groups, Amer. J. Math., 134 (2012), pp. 1679– 1704. [20] P. Niroomand and M. Parvizi, 2-nilpotent multipliers of a direct product of Lie algebras, Rend. Circ. Mat. Palermo (2), 65 (2016), pp. 519–523.
[21] D. J. Saltman, Noether’s problem over an algebraically closed field, Invent. Math., 77 (1984), pp. 71–84.
[22] V. S. Varadarajan, Lie groups, Lie algebras, and their representations, vol. 102 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1984. Reprint of the 1974 edition.
[23] J.-B. Zuber, Invariances in physics and group theory, in Sophus Lie and Felix Klein: the Erlangen program and its impact in mathematics and physics, vol. 23 of IRMA Lect. Math. Theor. Phys., Eur. Math. Soc., Z¨urich, 2015, pp. 307–326.