Much of the recent progress in the representation theory of infinitesimal group
schemes rests on the application of algebro-geometric techniques related to the notion
of cohomological support varieties (cf. [6, 8–10]). The noncohomological characterization
of these varieties via the so-called rank varieties (see [21, 22]) involves
schemes of additive subgroups that are the infinitesimal counterparts of the
elementary abelian groups. In this note we introduce another geometric tool by
considering schemes of tori of restricted Lie algebras. Our interest in these derives
from the study of infinitesimal groups of tame representation type, whose
determination [12] necessitates the results to be presented in §4 and §5 as well as
techniques from abstract representation theory.In contrast to the classical case of complex Lie algebras, the information on the
structure of a restricted Lie algebra that can be extracted from its root systems is
highly sensitive to the choice of the underlying maximal torus. Schemes of tori obviate
this defect by allowing us to study algebraic families of root spaces. Accordingly,
these schemes may also shed new light on various aspects of the structure theory of
restricted Lie algebras. We intend to pursue these questions in a forthcoming paper
[13], and focus here on first applications within representation theory.
Lie Algebras, Structure of Nonlinear Systems and Chaotic Motion