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Course Syllabus: Click Here.
Weekly Schedule: Announcements about homework, special lectures, etc. will be posted here.
1. Week of August 1st.
Lecture 1 (01/08): Definition of a Lie group.
Examples: GL(n,R), GL(n,C), SL(n,R), SL(n,C), PGL(n,R), PGL(n,C), O(n), U(n).
Review of the topological properties of a space being (i) compact, (ii) connected.
A fun exercise in elementary number theory.
Galois groups: compact topological groups (but not Lie groups).
Lecture 2 (02/08): More examples: O(n), SO(n), U(n), SU(n), Sp(n), Lorentz group: SO(3,1).
(For us Sp(n) will be the compact symplectic group; Notation to be compared with Sp(2n,R) or Sp(2n,C).)
Maximal compact subgroups of general linear groups.
Special Lecture on Basics of Group Theory: August 3rd, Friday, 5:40 p.m. to 7:00 p.m., Madhava Hall.
Homework 1: Click Here. (Due Date: 8th August in class. Submit exercises 4 and 10 for grading.)
2. Week of August 6th.
Lecture 3 (06/08): Tangent vectors, and the tangent space at a point p of a manifold M.
Examples of tangent space computations: M = V, GL(n,R), SO(2).
Functoriality of tangent space: The Adjoint representation.
Lecture 4 (08/08): Review of Lecture 3.
Tangent Bundle of a manifold.
Vector fields on a manifold; the Lie bracket of two vector fields.
Lecture 5 (09/08): The Lie algebra of a Lie group.
The Adjoint representation of a Lie group.
Homework 2: Click Here. (Due Date: 15th August in class. Submit exercises 4 and 10 for grading.)
3. Week of August 13th.
Lecture 6 (13/08): The differential of determinant is trace.
Integral curves on manifolds and one-parameter groups of a Lie group.
The exponential of a matrix.
The Lie algebras of SL(n), O(n), SO(n), U(n), SU(n).
Lecture 7 (15/08): The Lie algebras of SO(n) and U(n).
Explicit examples of exponentials of matrices.
(Note: This is a special lecture on independence day. There won't be any lecture on 16th, as I am out of town.)
Homework 3: Click here. (Due Date: 23rd August in class.)
4. Week of August 20th.
Lecture 8 (20/08): The exponential map of a Lie group.
Properties: Smooth, locally diffeomorphic, and natural.
(The notes are as in Lecture 7.)
(No lecture on 22/08 being a holiday for Bakri Id.)
Lecture 9 (23/08): Classification results for: connected abelian Lie group, and a compact abelian Lie group.
Homework 4: Click here. (Due date: 29th August in class.)
5. Week of August 27th.
Lecture 10 (27/08): Lie subgroups. Submanifolds. Closed subgroup theorem. A continuous homomorphism between Lie groups is smooth.
Lecture 11 (29/08): Homogeneous spaces. Quotient groups. If N is a closed normal subgroup of a Lie group G then G/N is a Lie group.
Lecture 12 (30/08): Examples: SL(2,R)/SO(2), GL(2,R)/B(2,R), and SO(n)/SO(n-1).
Homework 5: Click Here. (Due date: September 3rd in class.)
6. Week of September 3rd.
Lecture 13 (03/09): Haar integrals.
Lecture 14 (05/09): Examples of Haar integrals.
Lecture 15 (06/09): Basic definitions of representation theory.
Homework 6: Click here. (Due Date: September 17th, Monday, in class.)
7. Week of September 10th.
Lecture 16 (10/09): Complete reducibility theorem. Schur's Lemma.
Note: there will be no more lectures this week (I am out of town on Wednesday (12th) and Thursday (13th); but 13th is a holiday anyway).
8. Week of September 17th.
Lecture 17 (17/09): Multilinear algebra.
Lecture 18 (19/09): Basic character theory; characters of direct sums, duals, tensor product, symmetric and exterior square.
Lecture 19 (20/09): Characters = degree 1 representations; commutator subgroup and abelianization; all the characters of GL(2,R).
Homework 7: Click here.
Midterm Exam: Friday, September 28th, 3:00 to 5:00 p.m., in LHC 203.
Syllabus for midterm exam: Based on Homework sets 1 through 7; and up to Lecture 19.
One A4 size paper per student with hand-written notes (on both sides) will be allowed.
Some Announcements:
Last Lecture for this course will be on November 12th.
Special lectures on these Fridays: October 5th, October 12th, October 26th, November 9th.
All these special lectures will be from 5:30 to 6:30 pm in Madhava Hall.
9. Week of October 1st.
No lecture on October 3rd as I am of town.
Lecture 20 (04/10): Schur's orthogonality relations; orthonormality of characters; irreducibility criterion;
Example: the standard representation of S_4 is irreducible. (This is true for S_n.)
Lecture 21 (05/10): The character of a permutation representation; the regular representation of a finite group; Burnside's formula.
Example: The dimensions of irreducible representation of S_4 are 1, 1, 3, 3, 2.
10. Week of October 8th.
Lecture 22 (08/10): Proof of Schur's orthogonality relations; Regular representation; Bases for Class functions.
Lecture 23 (10/10): Representations of G x H, Character Table of a finite group.
Homework 8: Click Here. (Due in class on Monday, October 15th.)
Lecture 24 (11/10): Character tables for S_3, S_4, A_4, S_4. Get started on Induced representations.
Lecture 25 (12/10): Induced representations and Frobenius Reciprocity.
Frobenius's paper which has induced representations and Frobenius reciprocity.
11. Week of October 15th.
Lecture 26 (15/10): Mackey Theory. Example: the irreducible representations of dihedral groups.
Homework 9: Click Here. (Due in class on Wednesday, October 24th.)
Festival Break from October 17 to October 21.
12. Week of October 22nd. (Representations of compact Lie groups SO(2), SU(2), U(2), SO(3).)
Lecture 27 (22/10): Characters of SO(2). Construction of irreducible representations of SU(2).
Lecture 28 (24/10): All irreducible representations of SU(2). Representations of U(2).
Lecture 29 (25/10): Irreducible representations of SO(3) on the space of harmonic homogeneous polynomials.
Lecture 30 (26/10): Completing the proofs of classification of irreducible representations of SO(2), SO(3), SU(2) and U(2).
Homework 10: Click Here. (Due in class on Wednesday, October 31st.)
13. Week of October 29th. (Peter Weyl Theory for a compact topological group G.)
Lecture 31 (29/10): The structure of the space spanned by matrix coefficients M(G) inside C(G).
Lecture 32 (31/10): Statement of Peter-Weyl theory: M(G) is dense in C(G), and L^2(G) as a representation of G x G.
Lecture 33 (01/11): Consequences of Peter-Weyl theory: There is a faithful representation. Every irreducible is finite-dimensional.
Homework 11: Click Here. (Due in class on Thursday, November 8th.)
14. Week of November 5th.
Lecture 34 (05/11): Clean-up Peter Weyl theory. The representation theory of the Lie algerbra sl(2).
Lecture 35 (08/11): The representation theory for a general compact Lie group via the example: U(n).
Root space decomposition, Root Systems, Weights of a representation, Highest weight.
"Highest Weight Theory": The highest weight knows everything of the irreducible representation that it parametrizes.
Weyl Character Formula. Weyl Dimension Formula.
Lecture 36 (09/11): Special Lecture by Professor Sunil Mukhi.
Title: The role of representation theory of compact Lie groups in Physics.
Venue: Madhava Hall. Time: 5:30 p.m.
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