Graduate Course: Introduction to Representations of Algebras

SEMESTER 1, 2009/10.

Lecturer: Robert Marsh
Office: 10.15 (Maths Satellite), Tel. 0113 343 5164.
Time: 10am-12, MALL 2.

Mondays 5, 12 October
Tuesdays 20, 27 October
Tuesdays 3, 17, 24 November
Tuesdays 1, 8 December.

An algebra is a vector space together with a bilinear multiplication. Abstract algebras can be studied via their representations, i.e. multiplication preserving linear maps to algebras of matrices, which make them in a sense more explicit.

Representations of algebras play a role in many different fields, for example in Lie theory, algebraic geometry, combinatorial geometry, linear algebra, noncommutative geometry, and differential equations. Quivers, or directed graphs, give a good way of studying both algebras and their representations. In particular, Auslander-Reiten theory provides a method for analysing the category of representations of an algebra by breaking down arbitrary morphisms into irreducible morphisms.

The course will be an introduction to representations of algebras from the perspective of quivers, and will include an introduction to Auslander-Reiten theory.

Two key textbooks covering this area are listed below. A good undergraduate background in algebra will be assumed but the course will be mainly self-contained.