By Bryant R.L.

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Principal Bundles. Let M be an n-manifold and let P be a principal right G-bundle over M. Thus, P comes equipped with a submersion π: P → M and a free right action ρ: P × G → P so that the ﬁbers of π are the G-orbits of ρ. The Gauge Group. , π◦φ=π and ρg ◦ φ = φ ◦ ρg for all g ∈ G. For reasons having to do with Physics, this group is nowadays referred to as the gauge group of P . Of course, Aut(P ) is not a ﬁnite dimensional Lie group, but it would have been considered by Lie himself as a perfectly reasonable “continuous transformation group” (although not a very interesting one for his purposes).

Center of G. 3 a b 0 1 a ∈ R+ , b ∈ R 35 is trivial, so any connected Lie group with the same Lie algebra is actually isomorphic to G. (In the next Lecture, we will show that whenever K is a closed normal subgroup of a Lie group G, the quotient group G/K can be given the structure of a Lie group. ) 20. Show that SL(2, R) is not a matrix group! In fact, show that any homomorphism φ: SL(2, R) → GL(n, R) factors through the projections SL(2, R) → SL(2, R). (Hint: Recall, from earlier exercises, that the inclusion map SL(2, R) → SL(2, C) induces the zero map on π1 since SL(2, C) is simply connected.

2. Let G be a Lie group and let H be an abstract subgroup. Show that if there is an open neighborhood U of e in G so that H ∩ U is a smooth embedded submanifold of G, then H is a Lie subgroup of G. 3. Show that SL(n, R) is an embedded Lie subgroup of GL(n, R). ) 4. Show that O(n) is an compact Lie subgroup of GL(n, R). (Hint: O(n) = F −1 (In ), where F is the map from GL(n, R) to the vector space of n-by-n symmetric matrices given by F (A) = t A A. Taking note of Exercise 2, show that the Implicit Function Theorem applies.