Module manager: Ilaria Colazzo
Email: I.Colazzo@leeds.ac.uk
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2026/27
MATH1120 Introduction to Group Theory, MATH2130 Further Linear Algebra and Discrete Mathematics AND MATH2120 Rings and Polynomials or equivalent
| MATH1120 | Introduction to Group Theory |
| MATH2120 | Rings and Polynomials |
| MATH2130 | Further Linear Algebra and Discrete Mathematics |
MATH3193 – Algebras and representations
This module is not approved as an Elective
In this module you will study associative algebras, vector spaces equipped with a compatible multiplication (for example, the algebra of complex square matrices). You will develop the core theory of algebras and their representations, leading up to Wedderburn’s classification of finite-dimensional semisimple algebras in terms of matrix algebras and the resulting structure of their modules. Along the way you will also meet basic categorical language, using notions such as functors and morphisms to express the theory in a modern, conceptual framework.
The aim of this module is to develop a rigorous understanding of associative algebras and their representations, with particular attention to a class of examples such as matrix algebras, group algebras and path algebras (for example, path algebras of quivers). You will learn to work with modules, homomorphisms and ideals, and to prove and use theorems such as the isomorphism theorems for algebras and modules, Wedderburn’s theorem and Maschke’s theorem in explicit classification problems. A further aim is to introduce basic categorical language, including categories, functors and morphisms, in order to express parts of representation theory in modern language.
Subject specific learning outcomes:
On successful completion of the module students will be able to:
Define and give examples of associative algebras, group algebras, path algebras, modules, submodules, ideals, homomorphisms, simple and semisimple modules and algebras.
Work out explicit examples of algebras and modules, for instance identifying ideals and submodules, forming quotients and writing modules as direct sums of simple modules when this is possible.
Prove key results about algebras and their representations, for example the isomorphism theorems for algebras and modules, and standard properties of simple and semisimple modules.
Understand and use fundamental theorems in the area, such as Schur’s lemma, Wedderburn’s structure theorem and Maschke’s theorem, to analyse and classify algebras and their representations.
Formulate and interpret basic constructions in representation theory using categorical language, by describing simple examples in terms of categories, functors and morphisms, and reading simple commutative diagrams.
Skills learning outcomes:
On successful completion of the module students will be able to:
Communicate complex ideas clearly in writing
Work independently to manage their learning, planning their time across the semester, meeting deadlines and acting on feedback from assessments and classes.
Analyse unfamiliar problems systematically.
Make use of digital tools to support their work where appropriate.
Collaborate constructively with peers, for example by discussing approaches to exercises.
Reflect on their own learning and problem-solving strategies, recognising strengths and areas for development.
Associative algebras and standard examples (e.g. matrix algebras, group algebras, path algebras); algebra homomorphisms and basic constructions.
Ideals and quotient algebras; kernels and images.
Modules over algebras: submodules, module homomorphisms, direct sums and quotients.
Basic categorical language for representation theory: categories, functors and morphisms; interpreting simple commutative diagrams.
Isomorphism theorems for algebras and modules.
Simple and semisimple modules and algebras; decomposition ideas; Schur’s lemma and consequences for endomorphisms of simple modules.
Wedderburn-type structure theory for finite-dimensional semisimple algebras; implications for the classification of modules/representations.
Additional topics that build on these may be covered as time allows. Such topics may be drawn from the following, or similar:
Maschke-type results and group representations.
Jacobson radical and semisimple quotients.
Further quiver/path algebra topics.
Projective modules and basic homological ideas.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 33 | 1 | 33 |
| Private study hours | 117 | ||
| Total Contact hours | 33 | ||
| Total hours (100hr per 10 credits) | 150 | ||
117
Formative feedback will be provided on regular example sets or other similar learning activity.
Check the module area in Minerva for your reading list
Last updated: 12/05/2026
Errors, omissions, failed links etc should be notified to the Catalogue Team