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Lie Groups and Lie Algebras
The course begins with an introduction to groups, covering basic properties, examples, and structures. We then introduce matrix Lie groups, focusing on key examples such as SO(n), SU(n), and SL(n,C). Next, we develop the general theory of Lie groups, leading to the exponential map and the definition of Lie algebras. The Baker-Campbell-Hausdorff formula is explored to understand how representations of a Lie group and a Lie algebra are related. We introduce basic representation theory, providing tools to analyze Lie groups and their actions. As a concrete example of a semisimple Lie group, we study SU(3), a local gauge symmetry for the strong force (described by QCD), followed by an examination of weights, roots, the Weyl group, and the Cartan subalgebra. This leads naturally to the classification of Lie algebras through root systems, weight diagrams, and Dynkin diagrams. Finally, we conclude with applications to quantum mechanics and symmetries in physical systems.1. Introduction to groups.2. Introduction to matrix Lie groups.3. Lie groups.4. The exponential map and Lie algebras.5. Baker-Campbell-Hausdorff formula.6. Introduction to representation theory.7. A study of SU(3) as an example of a semisimple Lie group.8. Weights, roots, the Weyl group and Cartan subalgebra.9. Root systems, weight and Dynkin diagrams.10. Applications to quantum mechanics and symmetries in physical systems.
Understand the fundamentals of group theory, including basic properties, examples, and structures of groups relevant to Lie theory.Define and analyse matrix Lie groups, recognizing key examples such as SO(n), SU(n), and SL(n,C).Explain the concept of Lie groups, their differentiable structure, and their significance in mathematics and physics.Apply the exponential map to relate Lie groups and Lie algebras and understand the structure of Lie algebras associated with matrix Lie groups.Interpret the Baker-Campbell-Hausdorff formula, demonstrating its role in connecting Lie group and Lie algebra representations.Gain a foundational understanding of representation theory, including how Lie groups and Lie algebras act on vector spaces.Analyze the structure of SU(3) as an example of a semisimple Lie group and understand its relevance in physics, particularly in quantum chromodynamics (QCD).Compute and interpret weights, roots, the Weyl group, and the Cartan subalgebra, applying these concepts to the classification of Lie algebras.Understand the classification of Lie algebras through root systems, weight diagrams, and Dynkin diagrams.
Subject to approval of the Head of School.
Students must attend one activity from each section.
Chris Stevens
Library portalLearn Postgraduate
Domestic fee $1,138.00
International Postgraduate fees
* All fees are inclusive of NZ GST or any equivalent overseas tax, and do not include any programme level discount or additional course-related expenses.
For further information see Mathematics and Statistics .