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Article Dans Une Revue Journal of Lie Theory Année : 2016

ON THE VARIETY OF FOUR DIMENSIONAL LIE ALGEBRAS

Laurent Manivel

Résumé

Lie algebras of dimension $n$ are defined by their structure constants , which can be seen as sets of $N = n^2 (n − 1)/2$ scalars (if we take into account the skew-symmetry condition) to which the Jacobi identity imposes certain quadratic conditions. Up to rescaling, we can consider such a set as a point in the projective space $P^{N−1}$. Suppose $n =4$, hence $N = 24$. Take a random subspace of dimension $12$ in $P^{23}$ , over the complex numbers. We prove that this subspace will contain exactly $1033$ points giving the structure constants of some four dimensional Lie algebras. Among those, $660$ will be isomorphic to $gl_2$ , $195$ will be the sum of two copies of the Lie algebra of one dimensional affine transformations, $121$ will have an abelian, three-dimensional derived algebra, and $57$ will have for derived algebra the three dimensional Heisenberg algebra. This answers a question of Kirillov and Neretin.
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Dates et versions

hal-01161618 , version 1 (08-06-2015)

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Laurent Manivel. ON THE VARIETY OF FOUR DIMENSIONAL LIE ALGEBRAS. Journal of Lie Theory, 2016, 26, pp.1-10. ⟨hal-01161618⟩
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