40 2. Basic notions of representation theory
Problem 2.16.2. Classify irreducible finite dimensional representa-
tions of the two-dimensional Lie algebra with basis X, Y and commu-
tation relation [X, Y ] = Y . Consider the cases of zero and positive
characteristic. Is the Lie theorem true in positive characteristic?
Problem 2.16.3. (Hard!) For any element x of a Lie algebra g
let ad(x) denote the operator g g, y [x, y]. Consider the Lie
algebra gn generated by two elements x, y with the defining relations
ad(x)2(y) = ad(y)n+1(x) = 0.
(a) Show that the Lie algebras g1, g2, g3 are finite dimensional
and find their dimensions.
(b) (Harder!) Show that the Lie algebra g4 has infinite dimension.
Construct explicitly a basis of this algebra.
Problem 2.16.4. Classify irreducible representations of the Lie al-
gebra sl(2) over an algebraically closed field k of characteristic p 2.
Problem 2.16.5. Let k be an algebraically closed field of character-
istic zero, and let q
k×,q
= ±1. The quantum enveloping algebra
Uq(sl(2)) is the algebra generated by e, f, K,
K−1
with relations
KeK−1
=
q2e, KfK−1
=
q−2f,
[e, f] =
K K−1
q q−1
(if you formally set K =
qh,
you’ll see that this algebra, in an appro-
priate sense, “degenerates” to U(sl(2)) as q 1). Classify irreducible
representations of Uq(sl(2)). Consider separately the cases of q being
a root of unity and q not being a root of unity.
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