# LIE ALGEBRAS - 2021/2

Module code: MATM011

## Module Overview

This module develops students’ appreciation of algebraic structure through a study of Lie algebras and their matrix representations.

### Module provider

Mathematics

FISHER David (Maths)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 117

Lecture Hours: 33

Semester 1

## Prerequisites / Co-requisites

MAT1034 Linear Algebra

## Module content

Indicative content includes:

• Lie groups as a motivation for Lie algebras

• Lie algebras, subalgebras, ideals, quotient algebras, direct sums

• Derivations, homomorphisms and automorphisms of Lie algebras

• Representations of Lie algebras.

• Nilpotency, solvability, simplicity and semisimplicity.

• Engel’s Theorem and Lie’s Theorem.

• The Killing form. Cartan’s criteria.

• The Levi decomposition.

## Assessment pattern

Assessment type Unit of assessment Weighting
Examination EXAMINATION 75
School-timetabled exam/test IN-SEMESTER TEST (50 MINS) 10
Coursework COURSEWORK ASSIGNMENTS 15

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate their ability to

·         construct and interpret mathematical arguments in the context of this module;

·         display subject knowledge by recalling key definitions and results;

·         apply the techniques learnt to both routine and unfamiliar problems.

Thus, the summative assessment for this module consists of:

·         One two-hour examination at the end of Semester 2, worth 75% of the module mark.

·         In-semester  test, worth 10% of the module mark

·         Take-home assignment, worth 15% of the module mark.

Formative assessment and feedback

Students receive written comments on their marked coursework assignments.  Verbal feedback is provided in lectures and office hours.

## Module aims

• enhance students' appreciation of abstract algebraic structure theory,
• develop an understanding of Lie algebras and related concepts,
• provide a foundation for independent study of algebraic topics.

## Learning outcomes

 Attributes Developed 1 Know the definitions and properties of Lie algebras, subalgebras, ideals, homomorphisms and automorphisims K 2 Be familiar with standard examples of Lie algebras and their representations by matrices KC 3 Appreciate the relationships between the structures studied in this module and those encountered in previous algebraic modules KCT 4 Understand the concepts of nilpotent, solvable, semisimple and simple Lie algebras, and know criteria for these properties KC 5 Devise solutions of algebraic problems involving the concepts studied KC 6 Read and interpret varying presentations of the material, e.g. in research papers, CT 7 Construct simple proofs similar to those encountered in the module KCT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to provide:

• An enhanced awareness of the theory and applications of abstract algebra

• Experience of the methods used to interpret, understand and solve problems in the topics covered.

The learning and teaching methods include:

• Three 50-minute lectures per week for eleven weeks, some being used as tutorials, problem classes and a in-semester test.

• Online notes supplemented by additional examples in lectures.

• Unassessed coursework consisting of exercises from the notes.

• Personal assistance given to individuals and small groups in office hours.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.