Differential geometry and Lie groups for Physicists - what is where in video-lectures ==================================================== Lecture 1 Dual space, dual basis, transformation of bases and components 0:00:00 Organizational information 0:03:25 What is Mathematical Physics (and which part will be here) 0:07:50 Raoul Bott on linear algebra (quotation) 0:16:40 Change-of-basis matrix 0:27:40 Dual space L^* 0:35:55 Dual basis (in L^*) 1:00:40 Bilinear (canonical) pairing 1:08:45 The dual basis for a new basis in L 1:17:50 Transformation of components of vectors and co-vectors ............................................................................................................ Lecture 2 Tensors as maps, tensor product, decomposition w.r.t. basis 0:02:30 The second dual of L (canonically isomorphic to L) 0:18:05 Multilinear map 0:26:00 Tensors as multilinear maps 0:29:55 Components of a tensor 0:34:15 Transformation of components of a tensor 0:48:30 Several roles of the same tensor 0:51:45 A short note on fundamentalism 0:53:15 Tensor operations: Linear combination and contraction 1:13:55 Tensor operations: Tensor product 1:22:20 A basis of tensors, decomposition of a tensor w.r.t. the basis ............................................................................................................ Lecture 3 Metric and cometric, structured set, standard matrix groups 0:00:40 Metric tensor 0:08:45 The cometric 0:21:10 Lowering and raising of indices 0:37:30 Definition of a group 0:40:30 Group of bijections of a set 0:43:55 Bijections of structured sets 0:50:00 Linear structure and the group Aut V = GL(V) 0:57:20 The group GL(n,R) and its isomorphisms with GL(V) 1:07:50 Adding a bilinear form 1:17:40 Symmetric non-degenerate case, groups O(r,s) and O(n) 1:25:20 Volume in V and SL(V), SL(n,R), SO(r,s) and SO(n) ............................................................................................................ Lecture 4 Affine, Euclidean and Poincaré groups as matrix groups 0:08:30 Hermitian scalar product and the groups U(n) and SU(n) 0:18:00 Affine group 0:24:40 Euclidean and Poincare groups 0:30:20 Affine (Euclidean, Poincare) group as matrix groups 0:40:00 Direct product G_1 x G_2 of groups 0:55:00 GL_+(n,R) = SL(n,R) x GL(1,R) 1:11:20 Rough idea of a manifold ............................................................................................................ Lecture 5 Manifolds and smooth maps, atlas for S^n and RP^n 0:00:30 Manifold - chart (local coordinates), smooth atlas 0:10:00 Angles as coordinates on spheres 0:20:40 Stereographic projection - other atlas (coordinates) on spheres 0:34:00 Projective space RP^n - definition and atlas 0:48:05 CP^n, atlas, complex manifold 0:52:50 Cartesian product of manifolds 0:58:25 Smooth maps of manifolds 1:09:35 Diffeomorphism, embedding, submanifold ............................................................................................................ Lecture 6 Surfaces, curve and function, Lie group 0:00:15 Immersion, embedding, submanifold (again, a bit more) 0:08:00 Whitney theorem 0:13:00 Surfaces in Cartesian spaces (constraints, parametrization) 0:33:00 Hypersurfaces 0:38:45 SL(n,R) as a hypersurface 0:44:20 A curve and a function on a manifold 0:56:35 The concept of a Lie group 1:17:55 Lie algebras (first steps, no connection with groups) ............................................................................................................ Lecture 7 Lie algebra (general), the Lie algebra of a (matrix) Lie group 0:01:10 Lie algebras - still with no connection to groups 0:05:45 Lie algebras from associative algebras 0:20:15 Poisson bracket as a commutator in a Lie algebra 0:27:30 Structure constants 0:46:15 The Lie algebra of a Lie group 1:05:10 Standard notations 1:07:10 Computation of Lie algebras gl(n,R), o(n), u(n), sl(n,R) ............................................................................................................ Lecture 8 1-parameter subgroups (and examples), exponential mapping 0:00:05 1-parameter subgroups 0:08:35 The differential equation for the 1-parameter subgroup 0:12:30 Solution of the equation (how any 1-parameter subgroup looks like) 0:23:20 1-parameter subgroups on SO(2) 0:33:45 Exponential mapping, exponential Lie groups 0:38:15 Exponential mapping for O(2) 0:49:30 1-parameter subgroups on SU(2) 1:12:10 1-parameter subgroups on SO(3) 1:24:30 1-parameter subgroup as a geodesic ............................................................................................................ Lecture 9 Derived homomorphism, representation of a group and Lie algebra 0:00:05 mapping of 1-parameter subgroup w.r.t. a homomorphism 0:07:20 The corresponding commutative diagram for f and derived f' 0:11:40 Algorithm for computing f' from (known) f 0:17:20 f' is a Lie algebra homomorphism 0:23:15 Example: f = det, f' = Tr 0:36:40 Representations of groups - basics 0:41:30 Representation rho = id (A |--> A, for matrix groups) 0:44:50 Examples: rho(A) =1, rho(A) = det A, rho(A)B=ABA^{-1} 0:53:00 Derived representation rho' (of the Lie algebra) 1:02:35 Example: For rho(A)B=ABA^{-1} we get rho'(C)B=CB-BC 1:11:35 Conjugation in a group G and its properties 1:18:50 The origin of Ad and ad representations ............................................................................................................ Lecture 10 Irreducible representation, intertwining operator, Schur's lemma 0:00:05 Recapitulation: conjugation, representations, Ad and ad 0:09:15 Explicit formulas how Ad and ad representations operate 0:18:35 Some useful notations for matrices of representations 0:23:10 Generator of a representation, in particular for Ad 0:31:10 Isomorphism of Lie algebras (su(2) and so(3) are such) 0:40:30 Reducible and irreducible representations 0:53:00 Example: rho(A)=A is irreducible for SO(2) 1:05:55 Intertwining operator (equivariant mapping) 1:13:10 Equivalent representations and (part of) Schur's lemmas ............................................................................................................ Lecture 11 Killing-Cartan form, tensor product of representations 0:00:01 Correction of a small mistake in Lecture 10 0:02:10 The Schur's lemma finished 0:16:10 What do all irreducible representations of U(1) look like 0:27:25 Invariant scalar product 0:45:50 Killing-Cartan form, its Ad-invariance, semi-simple Lie algebra 1:08:00 Tensor product of vector spaces 1:18:35 Tensor product of operators 1:23:10 Tensor product of representations of a group 1:26:45 Its derived representation (of the Lie algebra)