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## MATH3193 Algebras and Representations

### 15 creditsClass Size: 30

Module manager: Dr Alison Parker
Email: A.E.Parker@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2017/18

### 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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

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