Versal Deformations and Versality in Central Extensions of Jacobi's Schemes

2010/09/03 by Roger Carles, Carles, Roger, Toukaiddine Petit +1
Mathematics · Physics and Astronomy · #16 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1009.0696

openalex publication_date 2010/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ł_m be the scheme of the laws defined by the Jacobi's identities on \K^m with \K a field. A deformation of \g∈Ł_m, parametrized by a local \K-algebra \A, is a local \K-algebra morphism from the local ring of Ł_m at ϕ_m to \A. The problem to classify all the deformation equivalence classes of a Lie algebra with given base is solved by "versal" deformations. First, we give an algorithm for computing versal deformations. Second, we prove there is a bijection between the deformation equivalence classes of an algebraic Lie algebra ϕ_m=R\ltimesϕ_n in Ł_m and its nilpotent radical ϕ_n in the R-invariant scheme Ł_n^R with reductive part R, under some conditions. So the versal deformations of ϕ_m in Ł_m is deduced to those of ϕ_n in Ł_n^R, which is a more simple problem. Third, we study versality in central extensions of Lie algebras. Finally, we calculate versal deformations of some Lie algebras.

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