Lecture notes for Introduction to Representation Theory

1 Basic notions of representation theory
1.1 What is representation theory?
In technical terms, representation theory studies representations of associative algebras. Its general
content can be very briefly summarized as follows.
An associative algebra over a field k is a vector space A over k equipped with an associative
bilinear multiplication a, b ⊃ ab, a, b A. We will always consider associative algebras with unit,
i.e., with an element 1 such that 1 · a = a · 1 = a for all a A. A basic example of an associative
algebra is the algebra EndV of linear operators from a vector space V to itself. Other important
examples include algebras defined by generators and relations, such as group algebras and universal
enveloping algebras of Lie algebras.
A representation of an associative algebra A (also called a left A-module) is a vector space
V equipped with a homomorphism δ : A ⊃ EndV , i.e., a linear map preserving the multiplication
and unit.
A subrepresentation of a representation V is a subspace U → V which is invariant under all
operators δ(a), a A. Also, if V 1 , V 2 are two representations of A then the direct sum V 1 V 2
has an obvious structure of a representation of A.
A nonzero representation V of A is said to be irreducible if its only subrepresentations are
0 and V itself, and indecomposable 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 representations.
As mentioned above, the algebra A is often given to us by generators and relations. For
example, the universal enveloping algebra U of the Lie algebra sl(2) is generated by h, e, f with
defining relations
he − eh = 2e, hf − fh = −2f, ef − fe = h. (1)
This means that the problem of finding, say, N-dimensional representations of A reduces to solving
a bunch of nonlinear algebraic equations with respect to a bunch of unknown N by N matrices,
for example system (1) with respect to unknown matrices h, e, f.
It is really striking that such, at first glance hopelessly complicated, systems of equations can
in fact be solved completely by methods of representation theory! For example, we will prove the
following theorem.
Theorem 1.1. Let k = C be the field of complex numbers. Then:
(i) The algebra U has exactly one irreducible representation V d 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 defined by the formulas
δ(h) = x − y , δ(e) = x , δ(f) = y .
x y y x
(ii) Any indecomposable finite dimensional representation of U is irreducible. That is, any finite
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