Lecture notes for Introduction to Representation Theory
Lecture notes for Introduction to Representation Theory
Lecture notes for Introduction to Representation Theory
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
1 Basic notions of representation theory<br />
1.1 What is representation theory?<br />
In technical terms, representation theory studies representations of associative algebras. Its general<br />
content can be very briefly summarized as follows.<br />
An associative algebra over a field k is a vec<strong>to</strong>r space A over k equipped with an associative<br />
bilinear multiplication a, b ⊃ ab, a, b A. We will always consider associative algebras with unit,<br />
i.e., with an element 1 such that 1 · a = a · 1 = a <strong>for</strong> all a A. A basic example of an associative<br />
algebra is the algebra EndV of linear opera<strong>to</strong>rs from a vec<strong>to</strong>r space V <strong>to</strong> itself. Other important<br />
examples include algebras defined by genera<strong>to</strong>rs and relations, such as group algebras and universal<br />
enveloping algebras of Lie algebras.<br />
A representation of an associative algebra A (also called a left A-module) is a vec<strong>to</strong>r space<br />
V equipped with a homomorphism δ : A ⊃ EndV , i.e., a linear map preserving the multiplication<br />
and unit.<br />
A subrepresentation of a representation V is a subspace U → V which is invariant under all<br />
opera<strong>to</strong>rs δ(a), a A. Also, if V 1 , V 2 are two representations of A then the direct sum V 1 V 2<br />
has an obvious structure of a representation of A.<br />
A nonzero representation V of A is said <strong>to</strong> be irreducible if its only subrepresentations are<br />
0 and V itself, and indecomposable if it cannot be written as a direct sum of two nonzero<br />
subrepresentations. Obviously, irreducible implies indecomposable, but not vice versa.<br />
Typical problems of representation theory are as follows:<br />
1. Classify irreducible representations of a given algebra A.<br />
2. Classify indecomposable representations of A.<br />
3. Do 1 and 2 restricting <strong>to</strong> finite dimensional representations.<br />
As mentioned above, the algebra A is often given <strong>to</strong> us by genera<strong>to</strong>rs and relations. For<br />
example, the universal enveloping algebra U of the Lie algebra sl(2) is generated by h, e, f with<br />
defining relations<br />
he − eh = 2e, hf − fh = −2f, ef − fe = h. (1)<br />
This means that the problem of finding, say, N-dimensional representations of A reduces <strong>to</strong> solving<br />
a bunch of nonlinear algebraic equations with respect <strong>to</strong> a bunch of unknown N by N matrices,<br />
<strong>for</strong> example system (1) with respect <strong>to</strong> unknown matrices h, e, f.<br />
It is really striking that such, at first glance hopelessly complicated, systems of equations can<br />
in fact be solved completely by methods of representation theory! For example, we will prove the<br />
following theorem.<br />
Theorem 1.1. Let k = C be the field of complex numbers. Then:<br />
(i) The algebra U has exactly one irreducible representation V d of each dimension, up <strong>to</strong> equivalence;<br />
this representation is realized in the space of homogeneous polynomials of two variables x, y<br />
of degree d − 1, and defined by the <strong>for</strong>mulas<br />
<br />
δ(h) = x − y , δ(e) = x , δ(f) = y .<br />
x y y x<br />
(ii) Any indecomposable finite dimensional representation of U is irreducible. That is, any finite<br />
5