MAT4343/MAT5148:
Representation Theory
(Instructor:
The course will start on Monday, May 3, 13:30 in B004.
Description
for the non-experts: The course will give an introduction to the
representation theory of finite groups and of some concrete finite-dimensional Lie
algebras. No prior knowledge of Lie algebras is required.
A representation of a group is
a group homomorphism of the group into the group of invertible linear
transformations of a vector space. To know all, or at least some
representations of a group is a central topic of pure mathematics. The reason
for this is that representations arise in many different, often unexpected
situations. For example, the most important problem of number theory (in my
view), the so-called Langlands program (http://en.wikipedia.org/wiki/Langlands_program),
has to do with group representations. But group representations also are
important for analysis, combinatorics, physics and of course algebra.
In the first part of the
course we will study representations of finite groups, like the group of
permutations of n letters, on a finite-dimensional complex vector space. In this case one can give a complete
description of all representations. The theory which we will describe is one of
the most beautiful theories in mathematics: All
reasonable questions have simple and elegant answers.
In the second part of the
course we will study representations of Lie algebras. Roughly speaking, Lie
algebras are approximations of certain groups, for example of the so-called Lie
groups (I am sorry, we will not delve into the representation theory of Lie
groups). An important example of a Lie algebra is the vector space of all
endomorphisms of a vector space with the commutator product. Analogously to the
case of groups, a representation of a Lie algebra is a Lie algebra homomorphism
into the Lie algebra of endomorphisms of a vector space.
We will study representations
of concrete Lie algebras on finite-dimensional complex vector spaces. For
example, we will consider the Lie algebra of trace-0-matrices and describe all
its representations on finite-dimensional complex vector spaces. We will do all
this without developing the structure theory of semisimple Lie algebra (after
all, this is done in the course MAT4142).
The course will be an
excellent preparation for students interested in writing a 4th-year
memoir or a M.Sc. thesis in representation theory (“represented” in this
department by Professors Neher, Nevins, Salmasian, Savage, and Zaynullin). But
it is also very appropriate and useful for anybody interested in pure
mathematics.
The
official course description: Complex-valued representations of finite and compact
groups. Character theory, orthogonality relations, group rings. Induced
representations. Additional topics chosen from the representations of Lie
groups and Lie algebras.
Prerequisites:
MAT2143
and MAT3143. A prior knowledge of Lie algebras is not required.
Textbook:
William
Fulton and Joe Harris, Representation
Theory – A First Course, Graduate Texts in Mathematics 129, Springer-Verlag 1991. We will cover approximately the sections
1-4. 6(?), 8-15.
Time
of lectures: Monday and Wednesday, 13:30-15:30,
B004.
Final
exam: Monday, July 19, 13-17, SMD 428.
Information:
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