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Journal of Lie Theory 33 (2023), No. 3, 703--712
Copyright Heldermann Verlag 2023



Four-Dimensional Lie Algebras Revisited

Laurent Manivel
Paul Sabatier University, Toulouse, France
laurent.manivel@math.cnrs.fr

Bernd Sturmfels
(1) Max-Planck-Institut MiS, Leipzig, Germany
(2) Dept. of Mathematics, University of California, Berkeley, U.S.A.
bernd@mis.mpg.de

Svala Sverrisdˇttir
Dept. of Mathematics, University of California, Berkeley, U.S.A.
svalasverris@berkeley.edu



The projective variety of Lie algebra structures on a 4-dimensional complex vector space has four irreducible components of dimension 11. We compute their prime ideals in the polynomial ring in 24 variables. By listing their degrees and Hilbert polynomials, we correct an earlier publication and we solve a problem raised by Kirillov and Neretin in 1987.

Keywords: Classification of Lie algebras, irreducible component, degree, Hilbert polynomial, resolution of singularities.

MSC: 14C17, 14M99, 17B05.

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