Reductive Homogeneous Space

In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space ''X'' together with a transitive action on ''X'' by a Lie group ''G'', which acts as the symmetry group of the geometry.

For background and motivation see the article on the Erlangen program.

Formal definition

A Klein geometry is a pair where ''G'' is a Lie group and ''H'' is a closed Lie subgroup of ''G'' such that the (left) coset space ''G''/''H'' is connected. The group ''G'' is called the principal group of the geometry and ''G''/''H'' is called the space of the geometry (or, by an abuse of terminology, simply the ''Klein geometry''). The space of a Klein geometry is a smooth manifold of dimension
:dim ''X'' = dim ''G'' − dim ''H''.

There is a natural smooth left action of ''G'' on ''X'' given by
:$g \cdot (aH) = (ga)H.$
Clearly, this action is transitive (take ), so that one may then regard ''X'' as a homogeneous space for the action of ''G''. The stabilizer of the identity coset is precisely the group ''H''.

Given any connected smooth manifold ''X'' and a smooth transitive action by a Lie group ''G'' on ''X'', we can construct an associated Klein geometry by fixing a basepoint ''x''₀ in ''X'' and letting ''H'' be the stabilizer subgroup of ''x''₀ in ''G''. The group ''H'' is necessarily a closed subgroup of ''G'' and ''X'' is naturally diffeomorphic to ''G''/''H''.

Two Klein geometries and are geometrically isomorphic if there is a Lie group isomorphism so that . In particular, if ''φ'' is conjugation by an element , we see that and are isomorphic. The Klein geometry associated to a homogeneous space ''X'' is then unique up to isomorphism (i.e. it is independent of the chosen basepoint ''x''₀).

Bundle description

Given a Lie group ''G'' and closed subgroup ''H'', there is natural right action of ''H'' on ''G'' given by right multiplication. This action is both free and proper. The orbits are simply the left cosets of ''H'' in ''G''. One concludes that ''G'' has the structure of a smooth principal ''H''-bundle over the left coset space ''G''/''H'':
:$H\to G\to G/H .$

Types of Klein geometries

Effective geometries

The action of ''G'' on need not be effective. The kernel of a Klein geometry is defined to be the kernel of the action of ''G'' on ''X''. It is given by
:$K = \.$
The kernel ''K'' may also be described as the core of ''H'' in ''G'' (i.e. the largest subgroup of ''H'' that is normal in ''G''). It is the group generated by all the normal subgroups of ''G'' that lie in ''H''.

A Klein geometry is said to be effective if and locally effective if ''K'' is discrete. If is a Klein geometry with kernel ''K'', then is an effective Klein geometry canonically associated to .

Geometrically oriented geometries

A Klein geometry is geometrically oriented if ''G'' is connected. (This does ''not'' imply that ''G''/''H'' is an oriented manifold). If ''H'' is connected it follows that ''G'' is also connected (this is because ''G''/''H'' is assumed to be connected, and is a fibration).

Given any Klein geometry , there is a geometrically oriented geometry canonically associated to with the same base space ''G''/''H''. This is the geometry where ''G''₀ is the identity component of ''G''. Note that .

Reductive geometries

A Klein geometry is said to be reductive and ''G''/''H'' a reductive homogeneous space if the Lie algebra $\mathfrak h$ of ''H'' has an ''H''-invariant complement in $\mathfrak g$.

Examples

In the following table, there is a description of the classical geometries, modeled as Klein geometries.

References

*{{cite book , author=R. W. Sharpe , title=Differential Geometry: Cartan's Generalization of Klein's Erlangen Program , publisher=Springer-Verlag , year=1997 , isbn=0-387-94732-9

Differential geometry
Lie groups
Homogeneous spaces