Syllabus
0. Introduction/Recall: Representations as presentation of groups ('tablaeux') by symmetries of space and as 'manifestation of' of symmetries o space in Classical mechanics (Hamiltonian systems) and Quantum Mechanics. 1. Recall of smooth manifolds: definition and examples. Definition of Lie group. Implicit function theorem for manifolds and Lie groups (without proofs). Proofs of the basic examples, i.e., that R^n, S^1, GL(n, F), SL(n, F), O(n, F), SO(n, F), U(n), SU(n), T^n (tori), F = R, C, are Lie groups. Dimensions of these manifolds. 2. (Borel, Radon and) Haar measures: definitions and basic examples. Haar measure for the symmetry group of the affine line and GL(n,R).
Weil theorem on Haar measure (without proof, proof eventually for Lie groups). Modular factor. 3. a) Representations of Lie groups: (closed) invariant subspace, irreducibility, (complete) reducibility, Schur lemma, representation of commutative groups, associated representations: Hilbert sum, dual, tensor product representations on Hilbert spaces. Representations of S^1, Z, and R^n. A non-topological version of Pontrjagin (self-)duality: "dual of dual of S1 is S1" and "dual of dual of Z is Z". Connection of Fourier coefficients to the Fourier transform. 3. b) Representations of compact Lie group: unitarization, complete reducibility, finite dimension of irreducible representations (without proof) a Peter--Weyl theorem (without proof):
Examples: C_n (cyclic), S_3, S_4 (permutation groups).
(Appendix: Overview of the algebraic theory of representations of Lie algebras - definition of Lie algebras, Cartan subalgebras, root, positive root, simple root, fundamental weight.
Cartan's theorem on the classification of irreducible representations of simple Lie algebras. Examples: sl(2,C) a sl(3,C).) 4. Examples of irreducible representations: Representations of SU(2), i.e., Spin(3) - double cover of SO(3), thus of the inner spin symmetries.
O(n) and harmonic polynomials, i.e., connected to orbital states in quantum mechanics.
(5. Special functions: Bessel and Legéndre functions. Special functions as matrix coefficients.) 6. Super-vector spaces, super-algebras, Lie super-algebras. Examples: Grassmann algebra, gl(m|n), sl(m|n), and eventually osp(m|n).
(Basic overview of Kac's classification of simple Lie super-algebras.) 7. Representations of the Heisenberg group: Stone--Neumann theorem, Schrödinger representation, and canonical commutation relations (CCR, canonical quantization) as differentiation of the Schroedinger representation. (Segal--Shale--Weil representation.)
(8. Representations of semi-direct products, Mackey theorem about the "little group" without proof.
Applications to the representation of the affine line symmetries group and the Poincaré group - semidirect product of the indefinite orthogonal groop O(1,3) and the abelian translational R^4, Wigner-type classification of the irreducible representation of the Poincare group - thus a classification of particles according to special relativistic quantum mechanics.)
Annotation
An advanced course of group theory for physicists. It is following to the basic course of mathematics for physicists.