Course
Archives Theoretical Statistics and Mathematics Unit | |||||||

Course: Lie Groups and Lie Algebra Time: Currently not offered Level: Postgraduate |
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Syllabus Past Exams Syllabus:
1.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].2.Lie algebras in general, Nilpotent, solvable, semisimple Lie algebra, ideals, Killing form, Lies and Engels theorem. Universal enveloping algebra and Poincare-Birkhoff-Witt Theorem (without proof). [6 weeks]. 3. 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 Liealgebras and their relation, Cartans criterion for semisimplicity. Statements and examples of Global Cartan decomposition, Root space decomposition; Iwasawa decomposition; Bruhat decomposition. [6 weeks]. Suggested Texts : 1. J.E. Humphreys: Introduction to Lie algebras and representation theory, GTM (9), Springer-Verlag (1972). 2. S.C. Bagchi, S. Madan, A. Sitaram and U.B. Tiwari: A first course on representation theory and linear Lie groups, University Press (2000). 3. Serge Lang: SL(2,R). GTM (105), Springer (1998). 4. W. Knapp: Representation theory of semisimple groups. An overview based on examples, Princeton Mathematical Series (36), Princeton University Press (2001). 5. 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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