I N T R O D U C T I O N
In the last couple of years there has been a renewed interest in modular Lie algebra theory gener-
ated by Block and Wilson's [Bl-W] classification of restricted simple Lie algebras. The classification
of restricted simple Lie algebras can be compared to Kac's [Kac] earlier classification of complex
simple irreducible graded Lie algebras of finite growth. In both situations one must consider two
classes of simple Lie algebras: the classical Lie algebras and the Lie algebras of Cartan type.
The classical Lie algebras, in the modular case, have been characterized by Mills and Seligman
[M-S]. These Lie algebras arise, as in the usual complex finite-dimensional situation, by considering
all simple Lie algebras for which there exist a non-degenerate quotient trace form. On the other
hand, the Lie algebras of Cartan type do not arise in this way, nor do they arise as the Lie algebra of
an algebraic group. The Lie algebras of Cartan type, in the modular setting, are finite-dimensional
analogs of the infinite-dimensional Lie algebras which are used in the classification of pseudogroups
of diffeomorphisms of a differentiable manifold. Kostrikin and Shafarevitch [Ko-S] have provided a
unified construction of these algebras which fall into four subcategories denoted by W, S, H and
K.
In the study of Lie algebras one often investigates the representation theory through the use of an
associative algebra called the universal enveloping algebra. Let (L,\p\) be a restricted Lie algebra
[Jac] over an algebraically closed field F of characteristic p and U(L) be its universal enveloping
algebra. Consider the ideal generated by the central elements x^
—xv
for x 6 U(L) and set U(L) =
U(L)/I. The algebra U(L) is now a finite-dimensional associative algebra with dim^ U(L) =
pdimF L Moreover, the unitary representations of U(L) are in one to one correspondence with
"restricted representations" of L. (ie. }: L \—• Ql(Y) s u c n ^ n a ^ t(x^) ^(^)p-)
l
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