VERSAL DEFORMATIONS AND VERSALITY IN CENTRAL EXTENSIONS OF JACOBI SCHEMES

摘要

Let L-m be the scheme of the laws defined by the Jacobi identities on K-m with K a field. A deformation of g is an element of L-m, parametrized by a local ring A, is a local morphism from the local ring of L-m at phi(m) to A. The problem of classifying 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 phi(m) = R (sic) phi(n) in L-m and its nilpotent radical phi(n) in the R-invariant scheme L-n(R) with reductive part R, under some conditions. So the versal deformations of phi(m) in L-m are deduced from those of phi(n) in L-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.