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) 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.)

A few lines below in (Klein 1872) 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.

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.

## 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.

gauge groupstabilizer subgrouplocal model spacelocal geometryglobal geometrydifferential cohomologyfirst order formulation of gravity
generalLie group/algebraic group $G$subgroup (monomorphism) $H \hookrightarrow G$quotient (“coset space”) $G/H$Klein geometryCartan geometryCartan connection
examplesEuclidean group $Iso(d)$rotation group $SO(d)$Cartesian space $\mathbb{R}^d$Euclidean geometryRiemannian geometryaffine connectionEuclidean gravity
Poincaré group $Iso(d-1,1)$Lorentz groupMinkowski space $\mathbb{R}^{d-1,1}$Lorentzian geometrypseudo-Riemannian geometryspin connectionEinstein gravity
super Poincaré groupspin groupsuper Minkowski spacetime $\mathbb{R}^{d-1,1\vert N}$Lorentzian supergeometrysupergeometrysuperconnectionsupergravity
linear algebraic groupparabolic subgroup/Borel subgroupflag varietyparabolic geometry
orthochronous Lorentz groupconformal geometryconformal connectionconformal gravity
generalsmooth 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.

## References

• Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872)

translation by M. W. Haskell, A comparative review of recent researches in geometry , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (retyped pdf, retyped pdf, scan of original)

• 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

Revised on December 17, 2014 22:56:59 by Urs Schreiber (127.0.0.1)