Moduli spaces of low dimensional real Lie algebras

Abstract

Moduli spaces of low dimensional complex Lie algebras have a unique stratification, determined by deformation theory, in which each stratum has the structure of a projective orbifold, in fact, it is given by an action of a finite group on complex projective space. For low dimensional real Lie algebras, the picture is similar, except that the strata are orbifolds given by the action of finite groups on spheres. We give a complete decomposition of the moduli spaces of real three and four dimensional Lie algebras and compare the structure of these moduli spaces to that of the corresponding complex Lie algebras.