UNIVERSITY OF KENT
Confirmation that this version of the module specification has been approved by the School
Learning and Teaching Committee:
…13th May 2015………………………………….(date)
MODULE SPECIFICATION
1.
Title of the module: Introduction to Lie Groups and Lie Algebras (MA603 and MA561)
2.
School or partner institution which will be responsible for management of the
module: School of Mathematics, Statistics and Actuarial Science
3.
Start date of the module: MA603 September 2015; MA561 October 2010 (revised version
start date September 2015)
4.
The number of students expected to take the module: 15 (H) and 10 (M)
5. Modules to be withdrawn on the introduction of this proposed module and
consultation with other relevant Schools and Faculties regarding the withdrawal:
MA561 is an existing M-level module. A new H-level version of this module, MA603, is
being created.
6.
The level of the module (e.g. Certificate [C], Intermediate [I], Honours [H] or
Postgraduate [M]): MA603 – H-level; MA561 - M-level
7.
The number of credits and the ECTS value which the module represents: 15 (7.5
ECTS)
8.
Which term(s) the module is to be taught in (or other teaching pattern): Autumn or
Spring term
9.
Prerequisite and co-requisite modules:
Pre-requisite module: MA553 Linear Algebra or equivalent
There are no co-requisite modules.
10. The programmes of study to which the module contributes:
For the H-level module: BSc (Hons) Mathematics (including programme with a year in
industry), BSc (Hons) Mathematics with a Foundation Year, MMath Mathematics, Graduate
Diploma in Mathematics.
For the M-level module: International MSc in Mathematics and its Applications, MSc in
Mathematics and its Applications, MMath Mathematics,
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For the MMath Mathematics and MMathStat Mathematics and Statistics programmes,
either the H-level or the M-level module may be taken, but not both.
11. The intended subject specific learning outcomes:
On successful completion of this module, H-level students will
(i)
be aware of the range of algebraic, geometric and analytic issues that the study of
Lie groups and Lie algebras entail, be able to reason confidently from algebraic
definitions such as ideals, bilinear forms, representations and root spaces, be able
to calculate confidently with basic constructions such as vector fields, Lie brackets,
exponentials, and adjoint representations, and be able to determine the Lie algebra
of a Lie group and in particular to understand its nature as a tangent space to the
group;
(ii)
have developed intuition for the structure of the main examples of Lie groups and
Lie algebras that arise in applications, including nonlinear Lie group actions;
(iii)
developed awareness of non-commutative phenomena;
(iv)
be aware of topics which are an important tool of research in many areas of
Mathematics, Physics and Chemistry.
(v)
have understood and be able to discuss the role played by Lie groups and algebras
in at least one application area in detail;
(vi)
have used the computer algebra package MAPLE to perform calculations in
specified Lie groups and Lie algebras.
On successful completion of this module, M-level students will also:
(vii)
have a systematic understanding of the algebraic, geometric and analytic issues that the
study of Lie algebras and Lie groups entail;
(viii)
have developed a comprehensive understanding of techniques for the application of Lie
algebra.
12. The intended generic learning outcomes
On successful completion of this module, H-level students will have:
(i)
an enhanced ability to reason and deduce confidently from given definitions and
constructions;
(ii)
matured in their problem formulating and solving skills;
(iii)
consolidated a wide variety of Calculus and Linear Algebra based skills;
(iv)
an enhanced ability to read independently and manage their time;
(v)
enhanced skills with mathematical software.
On successful completion of this module, M-level students will
(vi)
be able to reason confidently and critically when solving complex problems;
(vii)
have an enhanced ability to learn independently.
13. A synopsis of the curriculum
The joint curriculum will consist of the following (with possible applications that could be
discussed given in parentheses):
(i)
(ii)
(iii)
Introduction to Lie Groups. Basic examples. Matrix groups GL(n), SL(n), SO(n),
Sp(n). Double covers such as SU(2) → SO(3). Transformation groups. Statements
of Ado’s Theorem. Nilpotent, solvable and semi-simple Lie groups.
Representations of SU(2). (application to spherical harmonics)
Tensor representations, Clebsch-Gordan series for SU(2) (applications to quantum
angular momentum).
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(iv)
(v)
(vi)
Calculus on Lie groups. Left invariant vector fields. The Lie algebra of a Lie
group. The exponential map.
Introduction to Lie algebras. The Lie algebras gl(n), sl(n), so(n), sp(n). Nilpotent,
solvable and semi-simple Lie algebras. The adjoint action of a group on its Lie
algebra, and of a Lie algebra on itself. Derivations.
Representations of sl(2) (applications to projective geometry, integrable systems
such as the Toda lattice).
And, in addition, for M-level students:
(vii)
The Lie group G2
(viii) The Killing form and its use in the calculation of the solvable and semisimple parts
of a Lie algebra;
(ix)
The classification of semisimple Lie algebras;
(x)
Lie group actions.
Invariants of Lie group actions; differential invariants
(applications to solving invariant differential equations) OR Clifford algebras and
spin groups.
14. Indicative Reading List
Recommended for background reading:
K. Tapp, Matrix groups for undergraduates, Student Mathematical Library 29 American
Math.Society, 2005.
A. Fässler and E. Stiefel, Group Theoretical Methods and their applications, Boston,
Birkh•auser,1992.
Supplementary Texts:
A.W. Knapp, Lie groups beyond an introduction, Birkh•auser, Second edition, 2002.
R. Gilmore, Lie groups, Lie algebras, and some of their applications, New York, Wiley,
(1974)
M.L. Curtis, Matrix Groups, Springer Verlag, Second edition (1984).
N. Jacobson, Lie algebras, New York, Interscience Publishers, (1962).
J.G.F. Belinfante and B. Kolma, A survey of Lie groups and Lie algebras with
applications and computational methods, Philadelphia, Society for Industrial and Applied
Mathematics, (1972).
15. Learning and Teaching Methods, including the nature and number of contact hours
and the total study hours which will be expected of students, and how these relate to
achievement of the intended module learning outcomes
Number of contact hours: . 42-48, including approx. 36 lectures and 6-12 example
classes (1-hour). At M-level, the module may be offered as a directed reading course. In
such a case, the directed reading will be supported by a mix of lectures, tutorials and
enhanced office hours. The assessment methods described below will not change.
Number of independent learning hours: 102-108
Total study hours: 150
Contact hours comprise a mix of lecture and example classes.
Student study hours would be distributed between consolidation of lecture material, the
working of exercises on exercise sheets and assessed exercises, using Maple to obtain
results and to check their work, independent reading and exam preparation.
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The lectures and example classes will cover the learning outcomes in sections 11 and
12.
16. Assessment methods and how these relate to testing achievement of the intended
module learning outcomes
Assessment: For H-level students, the module is assessed on the basis of examination
(80%), coursework (10%) and an assignment based on independent reading (10%).
For M-level students, the module is assessed on the basis of examination (70%),
coursework (2x10%) and an assignment based on independent reading (10%).
Examination: A 3-hour written examination in the Summer term that consists of multi-part
questions requiring a mix of long and short answers to test at varying levels of
proficiency the learning outcomes 11(i)-(iii) and 12(i)-(iii). In addition, the examination for
M-level students will assess 11(vii),(viii) and 12(vi).
Coursework: This would normally consist of open-book assignments of unseen
problems, either to be worked by hand or to test the use of Maple to solve and display
relevant computations and graphs. The coursework will assess learning outcomes
11(i),(vi) and 12(v). Different open-book assignments will be set for M-level students,
which additionally will assess learning outcomes 11(vii),(viii) and 12(v),(vi).
Independent reading: An assignment based on exercises arising from material that
students study on their own, or alternatively, a short 10-page project, according to
student choice. The assignment/project will assess learning outcomes 11(iv)-(vi) and
12(iv),(v). In addition, for M-level students, the coursework will assess learning outcomes
11(viii) and 12(vi),(vii).
17. Implications for learning resources, including staff, library, IT and space
Staff: Convener and moderator
IT and Space: suitable rooms for lectures and/or computer classes (max of 3 per week)
18. The School recognises and has embedded the expectations of current disability
equality legislation, and supports students with a declared disability or special
educational need in its teaching. Within this module we will make reasonable
adjustments wherever necessary, including additional or substitute materials,
teaching modes or assessment methods for students who have declared and
discussed their learning support needs. Arrangements for students with declared
disabilities will be made on an individual basis, in consultation with the University’s
disability/dyslexia support service, and specialist support will be provided where
needed.
19. Campus(es) or Centre(s) where module will be delivered: Canterbury
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