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Transformation Groups
Out of Modern Differential Geometry for Physicists by C.J.Isham
Karim Osman
01.06.2019
University of Vienna
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Table of contents
1. Introduction
Basic definitions
Notions of group actions
2. Homogeneous Space Characterization Theorem
Examples
3. Literature
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Introduction
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Basic definitions
– transformation groups are essential in theoretical physics. Start withbasic definitions:
– Notation: aboves definition commonly written as
– Example: linear representation as a G−action, with the set being avector space
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Basic defintions
– if set is a differentiable manifold M and group G acting on M
(written as G ↷M) is a Lie group, tempting to restrict part ofPerm(M) involved with action to diffeomorphisms and give itdifferential structure
– Possible but unnecessarily complicated (inf. dim. topology). So,instead, define:
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Notions of group actions
– Let G ↷M and G ↷M ′. A map f ∶M ↦M ′ is equivariant iffollowing diagram commutes
– Kernel of a G-action: K = {g ∈ G ∣gp = p∀p ∈M}. An action iseffective if K = {e}.
– G -action is free if ∀p ∈M, {g ∈ G ∣gp = p} = {e} (every point of theset is moved away by G ∖ {e}). Alternative definition: ifhx = gx ⇒ h = g , the action is free.If p,q ∈M and G ↷M freely ⇒ either ∄ or ∃! g ∈ G ∶ gp = q
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Notions of group actions
– A G−action transitive if ∀p,q ∈M ∃g ∈ G ∶ gp = q.
– The orbit Op of a G -action through p ∈M is defined asOp = {q ∈M ∣∃g ∈ G ∶ gp = q}. (The set of points that can bereached with the group action).
– The stabilizer/little/isotropy group Gp at a point p ∈M of a groupaction is defined asGp = {g ∈ G ∣gp = p}
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On the notions of group actions
– kernel measures part of group that is not involved in group action
– a free action is always effective, but the converse is not true(example: faithful linear representation)
– to show transitivity, suffices to show for some p0 ∈M all of M can bereached with some g ∈ G . Then, for arbitrary p,q ∈ G , they can beconnected by first going to p0 with p0 = g−1p and then to q from p0
– linear representation is never transitive, since ∄g ∈ G ∶ g 0⃗ ≠ 0⃗
– if M = G/H with G being a Lie group, H a closed subgroup andG ↷M as γg(g ′H) = gg ′H, this action is transitive. Additionally,can be shown that G/H posses an analytical manifold structure.Action is not free since ∀h ∈ G ∶ h(eH) = eH
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Homogeneous SpaceCharacterization Theorem
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The theorem
– Also known under the main theorem for transitive group actions.Simply said: it states that any space M where a group G actstransitively is "effectively" of the form G/H for some H ⊂ G . That’sthe idea, now the real definition:
– If G is a Lie group and M is a differential manifold, is jp adiffeomorphism?
– Gp is a closed subgroup. Hence G/Gp has analytic manifold structureCan be shown that if M is locally compact and connected and G iscompact, jp is a diffeomorphism
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Examples
– The n-sphere Sn is diffeomorphic to O(n + 1)/O(n).
– For the set of real positive-definite n × n symmetric matrices Sn, thefollowing is true:
– This last relation as S3,1 ≅ GL+(4,R)/SO(3,1) is important forhandling Lorentzian geometries in four-dimensional spacetime.
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Literature
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Literature
All content and all pictures were taken out of "Isham, Chris J. Moderndifferential geometry for physicists. Vol. 61. World Scientific, 1999."
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