On the cohomology of Frobenius model Lie algebras

Abstract

Lie algebra model theory studies the closure O() of the Gln(C)-orbit in the variety Ln of Lie algebra laws of a law on Cn using nonstandard analysis. In this context, given a Lie algebra law , a contraction of is a law μ with μ 2 O(). On the other hand, a perturbation of in Ln is a μ 2 Ln such that the absolute value of the difference of the structure constants of and μ over a standard basis is smaller than any strictly positive real standard. Keeping this in mind, a Lie algebra g0 = (Cn, μ0) is called a model relative to a property (P) if any Lie algebra law μ satisfying (P)contracts to μ0 and any perturbation of μ0 satisfies (P). The property (P) studied in the article under review is the one to be Frobenius, i.e. the property that there exists a linear form ! 2 g on the 2n-dimensional Lie algebra g whose differential is symplectic, i.e. !n 6= 0. A family F of Lie algebras satisfying a property (P) is called a multiple model relative to (P) if any Lie algebra satisfying (P) contracts to a member of F and any perturbation of a member of F satisfies (P). M. Goze found in [C. R. Acad. Sci., Paris, S´er. I 293, 425–427 a multiple model for the above stated property (P). The authors of the present article compute first and second cohomology space of the Lie algebras in this family with respect to the adjoint representation. Furthermore, they compute the first obstruction for infinitesimal deformations to be prolonged which turns out to be zero.