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).
https://www.isibang.ac.in/~adean/infsys/database/MMath/LGLA.html
- Teacher: Maneesh Thakur