The u-algebra U(L) has many structural features similar to the group algebra FG for G a finite
group. Like the group algebra, U(L) has a Hopf algebra structure [Sw] and is a Frobenius algebra
[Be]. Furthermore, the u-algebra is a symmetric algebra if and only if tr(adx) = 0 for all x £ L [Sc],
which certainly holds if L is a restricted simple Lie algebra. According to Maschke's theorem FG
is semisimple if and only if p \\ G |. Hochschild [Ho] proved an analogous version of this theorem
for u-algebras: U(L) is semisimple if and only if L is a torus.
In the case of group algebras the p-group plays an important role in studying the representation
theory. With the u-algebra we can consider restricted Lie algebras such that x^ = 0 for some k
for all Vx G L. These will be called p-nilpotent Lie algebras [S-F]. These Lie algebras give rise to
the same representation theory as p-groups.
Theorem. IfL is a restricted Lie algebra then the following are equivalent:
(1) U(L) is indecomposable as right U(L) module.
(2) L is p-nilpotent.
(3) dimF U(L)/Rad U{L) = 1.
Humphreys [Hu-1] proved that for classical Lie algebras if n is the number of positive roots then
pn divides the dimension of each projective module. (The Steinberg module is the only projective
of degree pn.) The corollary of the above theorem, which is an analog to Dickson's theorem for
group algebras, generalizes this statement for arbitrary Lie algebras.
Corollary. Let L be a restricted Lie algebra and S a p-nilpotent subalgebra of L. If V is a
projective U(L) module then
pdlmFS \dimFV.
Even though p-nilpotent Lie algebras and p-groups give rise to the same representation theory,
the analogy between their roles within a larger algebra or group is not as well understood. For
a group algebra FG has finite representation type if and only if the p-Sylow subgroups of G are
cyclic. Pollack [Po] investigated the question of representation type for u-algebras. He found that
Previous Page Next Page