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Indian Institute of Science Education and Research Bhopal

MTH 411: Introduction to Lie Groups and Lie Algebras (4)

Pre-requisites: Required : MTH 311, Desirable : MTH 301 Groups and Rings

Learning Objectives:

The proposed course aims at providing and introduction to Lie groups, Lie algebras and their representations. The first part of the course focuses on matrix Lie groups (closed subgroups of GL(n; C)) and Lie algebras. The second part of the course deals with representations of semisimple Lie groups and Lie algebras. We begin with SU(2) and SU(3), as these cases very well illustrate the ideas of Cartan subalgebras, the roots, weights and the Wey1 group. We also look at Semisimple Lie groups and Lie algebras in general towards the end.

Course Contents:

Matrix Lie Groups: Definition and examples; Lie group homomorphisms and isomorphisms, Lie subgroups, polar decomposition.

Lie algebras: matrix exponential and matrix logarithm (4), one parameter subgroups, the Lie algebra of a matrix Lie group, Lie subalgebras, complexification of a real Lie algebra

Banker-Campbell-Hausborff formula : Definition, computations on Heisenberg group, Integral form of B-C-H formula (no proof), the series form of the B-C-H formula (no proof), applications to exponential map

Representations of SU(3): Weights and roots, theorem of the highest weight, the Wey1 group, weight diagrams

Semisimple Lie algebras: Complete reducibility, examples of reductive and semisimple Lie algebras, Cartan subalgebras, roots and root spaces, inner products of roots and co-roots, the Wey1 group, root systems, positive roots, the example of sl(n,C) in detail, uniqueness results.

Suggested Books:

• Hall, Brian Lie Groups, Lie Algebras, and Representations. Graduate Texts in Mathematics, Vol. 222, Springer Verlag, 2003.
• Rossmann, Wulf. Lie Groups: An Introduction through Linear Groups. Oxford Graduate Texts in Mathematics 5, Oxford University Press, 2002.
• Humphreys, James E. Introduction to Lie Alogebras and Representation Theory. Graduate Texts in Mathematics, Vol. 9, Springer, 1973.
• Baker, Andrew. Matrix Groups: An Introduction to Lie Group Theory. Springer Veriag, 2002.

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