Transformation Groups

Out of Modern Differential Geometry for Physicists by C.J.Isham

Karim Osman

01.06.2019

University of Vienna

Table of contents

1. Introduction

Basic definitions

Notions of group actions

2. Homogeneous Space Characterization Theorem

Examples

3. Literature

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Introduction

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

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

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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