Elementary subalgebras of Lie algebras

We initiate the investigation of the projective varieties E(r,g)E(r,g) of elementary subalgebras of dimension r of a (p  -restricted) Lie algebra gg for various r≥1r≥1. These varieties E(r,g)E(r,g) are the natural ambient varieties for generalized support varieties for restricted representations of gg. We identify these varieties in special cases, revealing their interesting and varied geometric structures. We also introduce invariants for a finite dimensional u(g)u(g)-module M  , the local (r,j)(r,j)-radical rank and local (r,j)(r,j)-socle rank, functions which are lower/upper semicontinuous on E(r,g)E(r,g). Examples are given of u(g)u(g)-modules for which some of these rank functions are constant.