2024/25 Undergraduate Module Catalogue

MATH3193 Algebras and Representations

15 creditsClass Size: 50

Module manager: Dr. Ilaria Colazzo
Email: I.Colazzo@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2024/25

Pre-requisite qualifications

MATH2025 or MATH2026, or equivalent.

This module is not approved as a discovery module

Module summary

An algebra is a ring which is simultaneously a vector space over a field in a compatible way. Thus, while elements of an algebra can be multiplied by a scalar or added together as in a vector space, they can also be multiplied together. The complex numbers, regarded as a vector space over the real numbers, are a simple example. There are many beautiful examples of algebras, including group algebras, which have a basis given by the elements of a group; quiver algebras, which arise from directed graphs; and Temperley-Lieb algebras which come from certain geometric diagrams in the plane.Semisimple algebras form an important class of algebras, and one of the highlights of the course is Wedderburn's beautiful Structure Theorem, which classifies the semisimple algebras. Even though semisimplicity is quite an abstract concept, it turns out that each semisimple algebra has an explicit description in terms of algebras of matrices. This perspective is taken further via the study of all representations (or modules) of an algebra, in which every element of the algebra is replaced by a matrix, giving an explicit model for the algebra.

Objectives

On completion of this module, students should be able to:

a) define some of the main concepts about associative algebras and representations.
b) state and prove some of the basic results about associative algebras and representations.
c) compute in various examples of algebras.
d) compute bases for some examples of algebras given by generators and relations.
e) use the isomorphism theorems to construct isomorphisms between modules
f) determine whether or not an algebra is semisimple.

Syllabus

- Associative algebras and examples
- Division algebras
- Algebras given by generators and relations
- Modules and the Isomorphism Theorems
- Simple and semisimple modules
- Semisimple algebras and the Wedderburn Structure Theorem

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	33	1.00	33.00
Private study hours			117.00
Total Contact hours			33.00
Total hours (100hr per 10 credits)			150.00

Private study

Studying and revising of course material.
Completing of assignments and assessments.

Opportunities for Formative Feedback

Regular example sheets.

Methods of assessment

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	2 hr 30 mins	100.00
Total percentage (Assessment Exams)		100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 30/09/2024

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