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Lie groups support classes

Week 3: Worked Examples of Lie groups using the implicit function theorem for manifolds. We tried U(n). Lie group structure on $SU(2) \times S^1, S^3$ and $U(2)$, and showed that diffeomorphic manifolds do not necessarily give rise to isomorphisms of Lie groups. Lie groups as group objects in the category of smooth manifolds with smooth maps.

Week 4: Review of manifolds theory (that the kernel of the differential of a manifold defined as in Week 1 gives the Lie algebra). Review of definition of Lie bracket via bracket on left invariant vector fields. Semi-direct product and split short exact sequences of groups (not nec. abelian). See also here for a nice introduction to semi-direct products.

Week 5: 1- Paramater subgroups, exponential map definition and exercises on surjectivity (Luke Peachey).

Week 6: Some useful exercises: redundancy of smoothness of inverse, computing the relation between Lie bracket when defined using right invariant extensions rather than left invariant extensions. Baker-Campbell-Hausdorff formula.

Week 7: Heisenberg group, connected topological groups generated by a neighbourhood of identity. Exp as a local diffeomorphism allows easy computation of Lie algebras without having to compete kernel of differential.

Week 8: The action of a Lie group on a manifold, which is smooth, proper and has trivial stabilisers gives rise to smooth structure on the set parametrising orbits and gives the projection map the structure of a smooth submersion. We used this to prove some results about homogeneous spaces. In particular how any smooth manifold with a smooth transitive $G$ action will be diffeomorphic to the smooth manifold $G/{G_p}$ with given $G$ action. There was a question on the proof that $G/H$ was a homogenous space or $G$ a closed subgroup. In particular why $G\times G/H \to G/H$ must be smooth? If one is unhappy with the argument in these notes, you could argue using this theorem here that $\pi \circ m$ is smooth and $id \times \pi$ is a surjective smooth submersion and so the resulting $\theta$ is smooth, I prefer this argument because then there really is no ambiguity. We also showed $SO(n)/SO(n-1)$ was diffeomorphic to $S^{n-1}$ using this theory by showing the action on $S^{n-1}$ of $SO(n)$ was transitive and discussed many other such applications such as $\mathbb{R}^n$ being diffeomorphic to $(\epsilon(n) \rtimes O(n)) / O(n)$ and that $\mathbb{C}P^n$ is diffeomorphic to $SU(n+1)/(SU(1)×U (n))$. A nice further example is the Grassmanian $Gr(n,m)$ diffeomorphic to $O(m)/(O(n) × O(m − n))$

Aside for the students also studying algebraic geometry: namely putting the structure of algebraic variety on the space $Orb_G(X)$ under the action of a linear algebraic group on an algebraic variety is a subtle question which gave rise to the study of geometric invariant theory. There are two notions of quotient here, categorical, (which satisfies the same universal property as the set theoretic quotient but with regular maps) and (semi)-geometric (which in some sense directly parametrises orbits) due to this technicality. In particular it can be shown that $Spec(k[X]^G)$ is the natural choice for the categorical quotient- but this defines the quotient in the category of affine schemes unless one can show $k[X]^G \subset k[X]$ is a finitely generated integral domain. Computing invariants itself can be a hard task. Furthermore, when the geometric quotient exists it coincides with the categorical one but there is no guarantee of existence of this. One can introduce the notion of stability and discard bad points and work in a Zariski open $U$ in order to find a way to try to introduce a geometric quotient $U \to U/G$ where the quotient map is again regular.

Week 9 N/A:

Week 10: Completely integrable/integrable/involutive distributions examples and Frobenius theorem. Definition of a foliation plus foliations arising from integrable distributions. Every lie sub algebra arises as the Lie algebra of a unique connected Lie subgroup- and my key bullet points on the ideas of the proof by constructing the lie subgroup as the leaf containing the identity of a specially chosen foliation.

As a supplement to these support classes, I will hold a revision session in the week prior to your exam where we can cover exam style problems.