Symmetric Space - General Definition

General Definition

Let G be a connected Lie group. Then a symmetric space for G is a homogeneous space G/H where the stabilizer H of a typical point is an open subgroup of the fixed point set of an involution σ of G. Thus σ is an automorphism of G with σ2 = id_G and H is an open subgroup of the set

Because H is open, it is a union of components of Gσ (including, of course, the identity component).

As an automorphism of G, σ fixes the identity element, and hence, by differentiating at the identity, it induces an automorphism of the Lie algebra of G, also denoted by σ, whose square is the identity. It follows that the eigenvalues of σ are ±1. The +1 eigenspace is the Lie algebra of H (since this is the Lie algebra of Gσ), and the -1 eigenspace will be denoted . Since σ is an automorphism of, this gives a direct sum decomposition

with

The first condition is automatic for any homogeneous space: it just says the infinitesimal stabilizer is a Lie subalgebra of . The second condition means that is an -invariant complement to in . Thus any symmetric space is a reductive homogeneous space, but there are many reductive homogeneous spaces which are not symmetric spaces. The key feature of symmetric spaces is the third condition that brackets into .

Conversely, given any Lie algebra with a direct sum decomposition satisfying these three conditions, the linear map σ, equal to the identity on and minus the identity on, is an involutive automorphism.