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.
