6 2. Basic notions of representation theory

A nonzero representation V of A is said to be irreducible if its

only subrepresentations are 0 and V itself, and it is said to be inde-

composable if it cannot be written as a direct sum of two nonzero

subrepresentations. Obviously, irreducible implies indecomposable,

but not vice versa.

Typical problems of representation theory are as follows:

(1) Classify irreducible representations of a given algebra A.

(2) Classify indecomposable representations of A.

(3) Do (1) and (2) restricting to finite dimensional representa-

tions.

As mentioned above, the algebra A is often given to us by gener-

ators and relations. For example, the universal enveloping algebra U

of the Lie algebra sl(2) is generated by h, e, f with defining relations

(2.1.1) he − eh = 2e, hf − fh = −2f, ef − fe = h.

This means that the problem of finding, say, N-dimensional represen-

tations of A reduces to solving a bunch of nonlinear algebraic equa-

tions with respect to a bunch of unknown N ×N matrices, for example

system (2.1.1) with respect to unknown matrices h, e, f.

It is really striking that such, at first glance hopelessly compli-

cated, systems of equations can in fact be solved completely by meth-

ods of representation theory! For example, we will prove the following

theorem.

Theorem 2.1.1. Let k = C be the field of complex numbers. Then:

(i) The algebra U has exactly one irreducible representation Vd of

each dimension, up to equivalence; this representation is realized in

the space of homogeneous polynomials of two variables x, y of degree

d − 1 and is defined by the formulas

ρ(h) = x

∂

∂x

− y

∂

∂y

, ρ(e) = x

∂

∂y

, ρ(f) = y

∂

∂x

.

(ii) Any indecomposable finite dimensional representation of U is

irreducible. That is, any finite dimensional representation of U is a

direct sum of irreducible representations.

As another example consider the representation theory of quivers.