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The theory of Lie algebras is fundamental to the study of groups of continuous symmetries acting on vector spaces, with applications to diverse areas including geometry, number theory and the theory of differential equations. Moreover, since quantum mechanical systems are described by Hilbert spaces acted on by continuous symmetries, Lie algebras and their representations are also fundamental to modern mathematical physics. This subject develops the basic theory in a way accessible to both pure mathematics and mathematical physics students, with an emphasis on examples. The main theorems are: the classification of complex semi-simple Lie algebras in terms of Cartan matrices and Dynkin diagrams, and the classification of finite-dimensional representations of these algebras in terms of highest weight theory.
Intended learning outcomes
On completion of this subject, students should be able to demonstrate:
- An understanding of the abstract theory of Lie algebras, and how they arise from Lie groups;
- The ability to analyse examples of semisimple Lie algebras using the language of roots and coroots;
- An understanding of the abstract theory of representations of Lie algebras;
- The ability to analyse examples of representations using the language of weights; and
- Facility with some basic applications of this theory to the study of symmetries in physical systems.
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- Problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- Analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- Collaborative skills: the ability to work in a team; and
- Time-management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 31 January 2024