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I have touched a bunch of the entries related to the Erlangen program, trying to polish a bit, adding more cross links and more references.
(This includes the entries stabilizer group, coset, Klein geometry, Cartan geometry and maybe more.)
I have added a few more paragraphs in the Idea-section at Erlangen program.
One question: on which page in Klein 1872 does he actually speak about quotient spaces $G/H$ (in some way or other)?
I find that surprisingly difficult to read. Don’t you? Shows what some good terminology and formalism can do for you.
He certainly seems to want to realise groups as subgroups of larger groups fixing some additional figure. I was taken back to the initial foray into Klein 2-geometry over 8 years ago, where we were wondering how 1-geometries could be related:
DC: Back on the level of Kleinian 1-geometry, what role could group homomorphisms play in the Erlangen program? I guess what’s already treated is group inclusions, e.g., projective transformations within affine transformations within Euclidean transformations. But is there scope for more interesting homomorphisms?
JB: Good point. Sure! Each group determines, or we could say “is”, a Klein geometry, but groups form a category - in fact a 2-category, since groups are categories - so we get a 2-category of Klein geometries.
Ignoring the 2-morphisms for a moment, though they’re interesting and important, let’s think about the morphisms: group homomorphisms, viewed as morphisms between Klein geometries.
The examples you mention are already very interesting, because they show how geometry is a unified subject, not just a bunch of isolated “geometries”. They show that including a little group in a big one can be seen as making a geometry “more flexible”, by adding new transformations.
But, there’s another way inclusions of groups show up: in Klein geometry, a “figure” is more or less given by the subgroup that stabilizes it. But, a subgroup is an inclusion of groups! So, the examples you listed of group inclusions can also be seen as “figures”! The most famous example is getting affine geometry by taking projective geometry and restricting to the transformations that stabilize a “point at infinity”, or “line at infinity”, or…
I hadn’t noticed this dual viewpoint, for some reason.
Anyway, some other examples come from outer automorphisms of groups: the outer automorphism of SL(n) is the “duality” that switches points and lines, and for Spin(8) one has 3! acting as outer automorphisms, called “triality”. For simple Lie groups one can read these off from the Dynkin diagram symmetries.
There are also inner automorphisms, which are just “changes of reference frame”. The difference between inner and outer automorphisms is nicely handled by the 2-categorical structure of Grp. (Ever figured out what a natural transformation between a functor betwen groups is?)
This leaves the epimorphisms of groups: quotient maps. When you mod out the Euclidean group by the translation subgroup, you get the rotation group. What does this short exact sequence mean for the corresponding 3 Klein geometries?
…
I find that surprisingly difficult to read. Don’t you?
Yes, I find it surprisingly verbose. That’s why I was asking if somebody could point me to the page where some allusion to $G/H$ might be made, because I didn’t quite have the energy to read every single sentence.
Of course I do see the text talk about subgroups, that’s not the problem. I am wondering though about the historical origin of the observation that the canonical $G$-space for which a presribed subgroup $H$ is the stabilizer of points is $G/H$. If we call this “Klein geometry”, then there must be some hint that Klein actually considered this. Or maybe we are being generous and regard this as obvious enough a consequence to attribute it to him? (But of course with $\sim 150$ years of hindsight, all of this is rather trivial…)
Regarding morphisms: since a Klein geometry is, equivalently, a homomorphism $H \to G$, then a morphism of Klein geometries is a diagram
$\array{ H_1 &\longrightarrow& H_2 \\ \downarrow && \downarrow \\ G_1 &\longrightarrow& G_2 }$He seems to be working with the idea that the ’space’, a $n$-dimensional manifold, is already given. Then there’s a group of transformations on it respecting certain features of the space. Perhaps he doesn’t see the need to reconstruct the points from the group as they’re already given.
What I mean is: where does he speak of that manifold being a homogeous space of the form $G/H$? Maybe he doesn’t?
I know that’s what you mean. I was just suggesting a reason why he doesn’t. But I haven’t worked through the whole paper.
Oh, I see, sorry.
Ah, I see it now at then end of section 5. He calls $G/H$ a “Körper” “body”:
Suppose in space some group or other, the principal group for instance, be given. Let us then select a single configuration, say a point, or a straight line, or even an ellipsoid, etc., and apply to it all the transformations of the principal group. We thus obtain an infinite manifoldness with a number of dimensions in general equal to the number of arbitrary parameters contained in the group, but reducing in special cases, namely, when the configuration originally selected has the property of being transformed into itself by an infinite number of the transformations of the group. Every manifoldness generated in this way may be called, with reference to the generating group, a body.If now we desire to base our investigations upon the group, selecting at the same time certain definite configurations as space-elements, and if we wish to represent uniformly things which are of like characteristics, we must evidently choose our space-elements in such a way that their manifoldness either is itself a body or can be decomposed into bodies.
I have added pointers to this appearance of $G/H$ in Klein’s text to Erlangen program – Idea and to Klein geometry – History.
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