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.