Advertisement

Geometriae Dedicata

, Volume 140, Issue 1, pp 137–144 | Cite as

On varieties parametrizing graded complex Lie algebras

  • T. Foth
  • M. Tvalavadze
Original Paper
  • 31 Downloads

Abstract

We study spaces parametrizing graded complex Lie algebras from geometric as well as algebraic point of view. If R is a finite-dimensional complex Lie algebra, which is graded by a finite abelian group of order n, then a graded contraction of R, denoted by \({R^{(\varepsilon)}}\), is defined by a complex n × n-matrix \({\varepsilon=(\varepsilon_{ij})}\), i, j = 1, . . . , n. In order for \({R^{(\varepsilon)}}\) to be a Lie algebra, \({\varepsilon_{ij}}\) should satisfy certain homogeneous equations. In turn, these equations determine a projective variety X R . We compute the first homology group of an irreducible component M of X R , under some assumptions on M. We look into algebraic properties of graded Lie algebras \({R^{(\varepsilon)}}\) where \({\varepsilon\in X_R}\).

Keywords

Lie algebras Gradings Homology groups 

Mathematics Subject Classification (2000)

17B70 32Q55 32Q15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bahturin, Y., Tvalavadze, M.: Group gradings on G 2. Commun. Algebra (to appear)Google Scholar
  2. 2.
    Bahturin Y., Zaicev M.: Group gradings on simple Lie algebras of type A. J. Lie Theory 16(4), 719–742 (2006)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bahturin Y., Sehgal S., Zaicev M.: Group gradings on associative algebras. J. Algebra 241(2), 677–698 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bahturin Y., Shestakov I., Zaicev M.: Gradings on simple Jordan and Lie algebras. J. Algebra 283(2), 849–868 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Carles R.: Variétés des algèbres de Lie de dimension inférieure ou égale à 7. (French.). C. R. Acad. Sci. Paris Ser. A–B 289(4), A263–A266 (1979)MathSciNetGoogle Scholar
  6. 6.
    Carles R., Diakité Y.: Sur les variétés d’ algèbres de Lie de dimension  ≤  7. (French.) [Varieties of Lie algebras of dimension  ≤  7]. J. Algebra 91(1), 53–63 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Carrell J., Sommese A.: Some topological aspects of \({{\mathbb {C}}^\ast}\) actions on compact Kaehler manifolds. Comment. Math. Helv. 54(4), 567–582 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Draper C., Martín C.: Gradings on g 2. Linear Algebra Appl. 418(1), 85–111 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gerstenhaber, M.: On the deformation of rings and algebras I, II, III. Ann. Math. 79, 59–103 (1964); Ann. Math. 84, 1–19 (1966); Ann. Math. 88, 1–34 (1968)Google Scholar
  10. 10.
    Jacobson N.: Lie Algebras. Dover, New York (1979)Google Scholar
  11. 11.
    Kirillov, A., Neretin, Yu.: The variety A n of n-dimensional Lie algebra structures. In: Some Problems in Modern Analysis. Fourteen papers translated from Russian. Am. Math. Soc. Transl. Ser. 2, vol 137, pp 21–30 (1987)Google Scholar
  12. 12.
    Kirwan F.: Intersection homology and torus actions. J. Am. Math. Soc. 1(2), 385–400 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Moody R., Patera J.: Discrete and continuous graded contractions of representations of Lie algebras. J. Phys. A 24(10), 2227–2257 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Neretin, Yu.: An estimate for the number of parameters defining an n-dimensional algebra (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 51(2), 306–318, 447 (1987); translation in Math. USSR-Izv. 30(2), 283–294 (1988)Google Scholar
  15. 15.
    Nijenhuis A., Richardson R.: Cohomology and deformations in graded Lie algebras. Bull. Am. Math. Soc. 72, 1–29 (1966)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Vinberg, E., Gorbatsevich, V., Onishchik, A.: Structure of Lie groups and Lie algebras (Russian). In: Current Problems in Mathematics. Fundamental Directions. Itogi Nauki i Tehniki, vol 41, pp 5–259. Akad. Nauk SSSR, VINITI, Moscow (1990)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Western OntarioLondonCanada
  2. 2.Department of Mathematics and StatisticsUniversity of SaskatchewanSaskatoonCanada

Personalised recommendations