group theory

Contents

Idea

The Erlangen program (in German, Erlanger Programm ) is a project, begun by Felix Klein at Erlangen in the 19th century (Klein 1872), to study geometry via symmetry groups of “geometric shapes”, hence from the perspective of group theory. The idea is to take the elementary building blocks of geometry to be not just Euclidean spaces but more generally homogeneous spaces $G/H$. These have then also been called Klein geometries .

In (Klein 1872, section 1) the theme of the program is described as follows:

Given a manifold and a transformation group acting on it, to investigate those properties of figures [Gebilde] on that manifold which are invariant under [all] transformations of that group.

In modern language this means to consider a group of homeomorphisms (diffeomorphisms) acting on a (smooth) manifold together with its stabilizer subgroup of any prescribed submanifold. (The concept of Lie group only emerged at that time, in fact Klein and Sophus Lie were in close contact, see Birkhoff-Bennett.)

The group of all such transformations

by which the geometric properties of configurations in space remain entirely unchanged

is called the “Hauptgruppe”, translated to “principal group”.

A few lines below in (Klein 1872, section 1) is the converse statement

Given a manifold, and a transformation group acting on it, to study its invariants.

Hence to find the figures which are left invariant by a given group action.

Then in (Klein 1872, end of section 5) it says:

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.

This means in modern language, that if $G$ is the given group acting on a given space $X$, and if $S \hookrightarrow X$ is a given subspace (a configuration), then the “body” (“Körper”) generated by this is the coset

$G/Stab_G(S)$

of $G$ by the stabilizer subgroup of that configuration. In the case that $S$ is a point we would now call this the orbit of $S$. See also there at Stabilizer of shapes – Klein geometry.

The text goes on to argue that spaces of this form $G/Stab_G(S)$ are of fundamental importance:

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.

Following this, such coset spaces $G/H$ have come to also be called Klein geometries.

When it was proposed, the Erlangen program served to unify various different kinds of geometry, discovered and studied at that time, into a common framwork. On the other hand, many kinds of geometries without global symmetries are not Klein geometries, notably Riemannian geometry is (in general) not. But a (pseudo) Riemannian manifold is locally (tangentially) modeled on Euclidean space (Minkowski spacetime) and this local model space is a Klein geometry. The generalization of Klein geometry to such local situations is Cartan geometry, see below.

In homotopy type theory

In homotopy type theory the idea of a group of symmetries preserving a figure inside the larger group of symmetries acting on what the figure is inscribed in is represented by any map of the form:

$B H \to B G$

The homotopy fiber of such a map is the Klein space $G/H$.

Refinements and generalizations

From local to global geometry – Cartan geometry

While many types of geometries (such as Riemannian geometry) are not in general Klein geometries, they are locally like Klein geometries. This generalization of Klein geometry is known as Cartan geometry.

local modelglobal geometry
Klein geometryCartan geometry
Klein 2-geometryCartan 2-geometry
higher Klein geometryhigher Cartan geometry

In physics terminology this corresponds to “locally gauging” the symmetry group. For instance for $H \hookrightarrow G$ the inclusion of the Lorentz group into the Poincare group, then the corresponding Klein geometry is just Minkowski spacetime, but a corresponding Cartan geometry is any pseudo-Riemannian manifold, i.e. a spacetime characterized in the first-order formulation of gravity.

geometric contextgauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
differential geometryLie group/algebraic group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$Klein geometryCartan geometryCartan connection
examplesEuclidean group $Iso(d)$rotation group $O(d)$Cartesian space $\mathbb{R}^d$Euclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group $Iso(d-1,1)$Lorentz group $O(d-1,1)$Minkowski spacetime $\mathbb{R}^{d-1,1}$Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
anti de Sitter group $O(d-1,2)$$O(d-1,1)$anti de Sitter spacetime $AdS^d$AdS gravity
de Sitter group $O(d,1)$$O(d-1,1)$de Sitter spacetime $dS^d$deSitter gravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
conformal group $O(d,t+1)$conformal parabolic subgroupMöbius space $S^{d,t}$conformal geometryconformal connectionconformal gravity
supergeometrysuper Lie group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$super Klein geometrysuper Cartan geometryCartan superconnection
examplessuper Poincaré groupspin groupsuper Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
super anti de Sitter groupsuper anti de Sitter spacetime
higher differential geometrysmooth 2-group $G$2-monomorphism $H \to G$homotopy quotient $G//H$Klein 2-geometryCartan 2-geometry
cohesive ∞-group∞-monomorphism (i.e. any homomorphism) $H \to G$homotopy quotient $G//H$ of ∞-actionhigher Klein geometryhigher Cartan geometryhigher Cartan connection
examplesextended super Minkowski spacetimeextended supergeometryhigher supergravity: type II, heterotic, 11d

Higher Klein geometry

Aspects of Klein geometry may be generalized from groups to groupoids and even categories or $\infty$-groupoids. See at higher Klein geometry.

Logicality and invariance

Logicians have attempted to demonstrate that specifically logical constructions are those invariant under the largest group of transformations, in the sense of the Erlangen program. See logicality and invariance.

• Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872) Mathematische Annalen volume 43, pages 63–100 1893 (doi:10.1007/BF01446615)

English translation by M. W. Haskell:

A comparative review of recent researches in geometry, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (euclid:1183407629, LaTeX version retyped by Nitin C. Rughoonauth: arXiv:0807.3161)

• Garrett Birkhoff, M. K. Bennett, Felix Klein and His “Erlanger Program” (pdf)

• Vladimir Kisil, Erlangen Programme at Large: An Overview (arXiv:1106.1686)

• Jeremy Gray, Felix Klein’s Erlangen programme, in Landmark Writings in Western Mathematics, ed. I. Grattan-Guinness, Elsevier, p. 544-552, 2005

• Ernst Cassirer, The concept of Group and The Theory of Perception, Philosophy and Phenomenological Research 5(1), pp. 1-36, 1944. A philosophical treatment of the Erlangan Program.