Course Archives Theoretical Statistics and Mathematics Unit

Course: Lie Groups and Lie Algebras Level: Postgraduate Time: Currently not offered

Syllabus Past Exams Syllabus: i) Linear Lie groups: the exponential map and the Lie algebra of linear Lie group, some calculus on a linear Lie group, invariant differential operators, finite dimensional representations of a linear Lie group and its Lie algebra. Examples of linear Lie group and their Lie algebras, e.g., Complex groups: GL(n, C), SL(n, C), SO(n, C), groups of real matrices in those complex groups: GL(n, R), SL(n, R), SO(n, R), Isometry groups of Hermitian forms SO(m, n), U(m, n), SU(m, n). Finite dimensional representations of su(2) and SU(2) and their connection. Exhaustion using the lie algebra su(2). [2 weeks] ii) Lie algebras in general, Nilpotent, solvable, semisimple Lie algebra, ideals, Killing form, Lies and Engels theorem. Universal enveloping algebra and PoincareBirkhoff-Witt Theorem (without proof). [6 weeks] iii) Semisimple Lie algebra and structure theory: Definition of Linear reductive and linear semisimple groups. Examples of Linear connected semisimple/ reductive Lie groups along with their Lie algebras (look back at 2 above and find out which are reductive/semisimple). Cartan involution and its differential at identity; Cartan decomposition g = k + p, examples of k and p for the groups discussed above. Definition of simple and semisimple Lie algebras and their relation, Cartans criterion for semisimplicity. Statements and examples of Global Cartan decomposition, Root space decomposition; Iwasawa decomposition; Bruhat decomposition. [6 weeks] iv) If time permits, one of the following topics: (i) A brief introduction to Harmonic Analysis on SL(2, R). (ii) Representations of Compact Lie Groups and Weyl Character Formula. (iii) Representations of Nilpotent Lie Groups. Suggested Texts : (a) J.E. Humphreys: Introduction to Lie algebras and representation theory, GTM (9), Springer-Verlag (1972). (b) S.C. Bagchi, S. Madan, A. Sitaram and U.B. Tiwari: A first course on representation theory and linear Lie groups, University Press (2000). (c) Serge Lang: SL(2, R). GTM (105), Springer (1998). (d) W. Knapp: Representation theory of semisimple groups. An overview based on examples, Princeton Mathematical Series (36), Princeton University Press (2001). (e) B.C. Hall, Lie Groups, Lie Algebras and Representations: An Elementary Introduction, Springer (Indian reprint 2004). Top of the page Past Exams

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