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The Bogomolov multiplier of Lie algebras

Year 2020, Volume: 49 Issue: 3, 1190 - 1205, 02.06.2020
https://doi.org/10.15672/hujms.455076

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.

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.
Year 2020, Volume: 49 Issue: 3, 1190 - 1205, 02.06.2020
https://doi.org/10.15672/hujms.455076

Abstract

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.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Zeinab Araghi Rostani This is me 0000-0002-4758-0212

Mohsen Parvizi 0000-0002-8133-5245

Peyman Niroomand 0000-0001-6411-4574

Publication Date June 2, 2020
Published in Issue Year 2020 Volume: 49 Issue: 3

Cite

APA Araghi Rostani, Z., Parvizi, M., & Niroomand, P. (2020). The Bogomolov multiplier of Lie algebras. Hacettepe Journal of Mathematics and Statistics, 49(3), 1190-1205. https://doi.org/10.15672/hujms.455076
AMA Araghi Rostani Z, Parvizi M, Niroomand P. The Bogomolov multiplier of Lie algebras. Hacettepe Journal of Mathematics and Statistics. June 2020;49(3):1190-1205. doi:10.15672/hujms.455076
Chicago Araghi Rostani, Zeinab, Mohsen Parvizi, and Peyman Niroomand. “The Bogomolov Multiplier of Lie Algebras”. Hacettepe Journal of Mathematics and Statistics 49, no. 3 (June 2020): 1190-1205. https://doi.org/10.15672/hujms.455076.
EndNote Araghi Rostani Z, Parvizi M, Niroomand P (June 1, 2020) The Bogomolov multiplier of Lie algebras. Hacettepe Journal of Mathematics and Statistics 49 3 1190–1205.
IEEE Z. Araghi Rostani, M. Parvizi, and P. Niroomand, “The Bogomolov multiplier of Lie algebras”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 3, pp. 1190–1205, 2020, doi: 10.15672/hujms.455076.
ISNAD Araghi Rostani, Zeinab et al. “The Bogomolov Multiplier of Lie Algebras”. Hacettepe Journal of Mathematics and Statistics 49/3 (June 2020), 1190-1205. https://doi.org/10.15672/hujms.455076.
JAMA Araghi Rostani Z, Parvizi M, Niroomand P. The Bogomolov multiplier of Lie algebras. Hacettepe Journal of Mathematics and Statistics. 2020;49:1190–1205.
MLA Araghi Rostani, Zeinab et al. “The Bogomolov Multiplier of Lie Algebras”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 3, 2020, pp. 1190-05, doi:10.15672/hujms.455076.
Vancouver Araghi Rostani Z, Parvizi M, Niroomand P. The Bogomolov multiplier of Lie algebras. Hacettepe Journal of Mathematics and Statistics. 2020;49(3):1190-205.