Geometriae Dedicata

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

On varieties parametrizing graded complex Lie algebras

  • T. FothEmail author
  • M. Tvalavadze
Original Paper


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 \({\varepsilon\in X_R}\).


Lie algebras Gradings Homology groups 

Mathematics Subject Classification (2000)

17B70 32Q55 32Q15 


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© 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

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