PROJECTIVE MODULES 3
even though s/(2) has a maximal p-nilpotent subalgebra of degree 1, the u-algebra is of infinite,
but tame representation type. Recently Pfautsch, Voigt, [Pf-V] and Rickard [Ri] have shown that
the other restricted simple Lie algebras are of wild representation type. The problem with drawing
these comparisons manifests itself in the induction-restriction procedure. With group algebras when
one "induces" then "restricts" one has a direct sum decomposition. In the case of u-algebras this
doesn't always happen and Pollack provides a sufficient condition for this to occur. One interesting
consequence of this, which we will see later in this paper, is that for u-algebras a projective module
can be obtained by inducing from a non-projective module. This phenomenon cannot occur with
group algebras.
In this paper I will attempt to draw upon the analogy between the representation theory of group
algebras and u-algebras in order to compare the representation theory between the u-algebras of
classical and "non-classical" Lie algebras. The representation theory for classical Lie algebras has
been extensively studied over the last 20 years. In 1971 Humphreys [Hu-1] wrote a paper which
provides a detailed account of how linkage between Verma modules holds in the modular setting
and its effect on the block structure of the u-algebra. However, the major contribution of his paper
was the first statement and proof of the BGG reciprocity principle: [P(A) : F(/i)] = [V"(/x) : L(X)]
where P(A) is the projective cover of the irreducible module L(X) and V(fi) is a Verma module.
This reciprocity law is similar to Brauer reciprocity for finite groups.
In Chapter 1 we present a general BGG reciprocity law for graded restricted Lie algebras. Since
every restricted simple Lie algebra is endowed with the structure of a graded restricted Lie algebra
the reciprocity law will hold for both classical Lie algebras and Lie algebras of Cartan type. The
techniques of Jantzen and Holmes will be used to prove the existence of filtrations by V^- modules.
Moreover, a new proof of the reciprocity law will be given which does not use an ordering of the
weights. A more general setting to this theory can be found in [Ho-N].
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