LIE ALGEBRAS - 2024/5

Module code: MATM011

Module Overview

This module develops students' understanding of abstract algebra through a study of Lie algebras and their matrix representations.

This module builds on material on abstract algebra and matrices from MAT1031 Algebra and MAT1034 Linear Algebra. It also complements the material in MAT2048 Groups & Rings and MAT3032 Advanced Algebra.

Module provider

Mathematics & Physics

Module Leader

ZELIK Sergey (Maths & Phys)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 7

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

Overall student workload

Independent Learning Hours: 69

Lecture Hours: 33

Guided Learning: 15

Captured Content: 33

Module Availability

Semester 1

Prerequisites / Co-requisites

None.

Module content

Indicative content includes: 



  • Lie groups and the associated Lie algebras. 


  • Lie algebras and subalgebras. Ideals. Direct sums. 


  • Homomorphisms, isomorphisms and automorphisms of Lie algebras. Derivations. Quotient algebras. 


  • Representations of Lie algebras. 


  • Nilpotency and 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
School-timetabled exam/test In-Semester Test (50 min) 20
Examination End-of-Semester Examination (2 hours) 80

Alternative Assessment

N/A

Assessment Strategy

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


  • Understanding of subject knowledge, and recall of key definitions, theorems and propositions in the theory of Lie algebras.

  • The ability to construct simple proofs similar to those in the module.

  • The ability to identify and use the appropriate methods to solve problems relating to Lie algebras.



Thus, the summative assessment for this module consists of:


  • One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1, 2 and 5.

  • A synoptic examination (2 hours), worth 80% of the module mark, corresponding to all Learning Outcomes 1 to 5.



Formative assessment

There are two formative unassessed courseworks over an eleven week period, designed to consolidate student learning. 

Feedback

Students will receive feedback on both the formative unassessed courseworks and the in-semester test. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal feedback in office hours.

Module aims

  • Extend students¿ knowledge of abstract algebra and provide students with an introduction to Lie algebras.
  • Develop students' understanding of rigorous proofs in the context of abstract algebra.

Learning outcomes

Attributes Developed
001 Students will understand the definitions and properties of Lie groups, and Lie algebras and subalgebras. Students will be able to provide and recognise standard examples. KC
002 Students will understand the definitions and properties of ideals, homomorphisms, isomorphisms and automorphisms of Lie algebras, and quotient algebras. Students will be able to construct examples of each. KC
003 Students will understand the concepts of nilpotent, solvable, simple and semisimple Lie algebras. KC
004 Students will understand and be able to construct representations of Lie algebras. KC
005 Students will be able to 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:


  • Introduce students to the theory of Lie algebras.

  • Provide students with experience of methods used to interpret, understand and solve problems relating to Lie algebras.



  The learning and teaching methods include:


  • Three one-hour lectures for eleven weeks, with module notes provided to complement the lectures. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.

  • Two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive feedback on these courseworks as guidance on their progress and understanding.

  • Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures.



 

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.

Reading list

https://readinglists.surrey.ac.uk
Upon accessing the reading list, please search for the module using the module code: MATM011

Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MATM011 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

Employability: The module MATM011 equips students with skills which significantly enhance their employability. The mathematical proficiency gained will hone their critical thinking and problem-solving abilities. Students will learn to evaluate complex algebraic problems, break them into manageable components, and apply the theory of Lie algebras and logical reasoning to arrive at solutions. These are highly sought after skills in many professions.

Global and Cultural Capabilities: Students enrolled in MATM011 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in lectures, which naturally facilitates the sharing of different cultures.

Resourcefulness and Resilience: MATM011 is a module which demands a rigorous approach to abstract algebra and the theory of Lie algebras, to which students will learn to adapt. They will gain skills in formulating rigorous mathematical proofs and analysing abstract algebraic problems using lateral thinking. Students will complete assessments which challenge them and build resilience.

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.