OverviewLie groups and their associated Lie algebras are studied by both pure and applied mathematicians and by physicists; this is a topic renowned for both its mathematical beauty and its immense utility. Lie groups include translation, rotation and scaling groups as well as unitary, symplectic and special linear matrix groups. We will study in detail the lower dimensional groups that arise in many applications, and more general theory such as the structure of their associated Lie algebras. Special topics include a look at the lowest dimensional exceptional Lie group G2, and Lie group actions and their invariants.
This module appears in:
Method of assessment
70% Examination, 30% Coursework
ML Curtis, Matrix Groups. (Springer Verlag, Second edition, 1984) (B)
R Gilmore, Lie groups, Lie algebras, and some of their applications. (New York, Wiley, 1974) (R)
N Jacobson, Lie algebras. (New York, Interscience Publishers, 1962) (B)
AW Knapp, Lie groups beyond an introduction. (Birkhäuser, Second edition, 2002) (B)
K Tapp, Matrix groups for undergraduates. (Student Mathematical Library 29, American Mathematical
Society, 2005) (R)
A Fässler & E Stiefel, Group Theoretical Methods and their applications. (Boston, Birkhäuser, 1992) (R)
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.