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2009/10 Taught Postgraduate Module Catalogue

MATH5193M Advanced Algebras and Representations

15 creditsClass Size: 30

Module manager: Dr A Hubery
Email: ahubery@maths.leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2009/10

Pre-requisite qualifications

MATH2033 and (MATH2080 or MATH2200).

This module is mutually exclusive with

MATH3193

Algebras and Representations

This module is approved as an Elective

Module summary

Multiplication by a complex number of absolute value one gives a rotation of the complex plane. In the 19th Century, Hamilton tried to find an algebra which would represent rotations of three-dimensional space. After much thought he realized that he had to give up commutativity and needed a four-dimensional algebra, his quaternions. Soon a beautiful theory had developed, with Grassmann and Clifford providing other examples of algebras (now used in quantum mechanics and relativity) and Wedderburn determining the structure of all 'semisimple' algebras. This module also includes an introduction to the notions of quivers and their representations, which provide a combinatorial language for the description of algebras with beautiful properties. As well as being of interest in its own right, this module should extend and consolidate the students' understanding of linear 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.
g) Determine whether or not a representation is indecomposable.

Syllabus

Associative algebras and examples. Division algebras. Algebras given by generators and relations. Modules and the Isomorphism Theorems. Simple and semisimple modules. Semisimple algebras. Representations of quivers.

Teaching methods

Delivery type	Number	Length hours	Student hours
Example Class	7	1.00	7.00
Lecture	26	1.00	26.00
Private study hours			117.00
Total Contact hours			33.00
Total hours (100hr per 10 credits)			150.00

Opportunities for Formative Feedback

Regular example sheets.

Methods of assessment

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	3 hr	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: 22/03/2010

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