Principal Homogeneous Space

In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial. Equivalently, a principal homogeneous space for a group G is a non-empty set X on which G acts freely and transitively, meaning that for any x, y in X there exists a unique g in G such that x·g = y where · denotes the (right) action of G on X. An analogous definition holds in other categories where, for example,

• G is a topological group, X is a topological space and the action is continuous,
• G is a Lie group, X is a smooth manifold and the action is smooth,
• G is an algebraic group, X is an algebraic variety and the action is regular.

If G is nonabelian then one must distinguish between left and right torsors according to whether the action is on the left or right. In this article, we will use right actions. To state the definition more explicitly, X is a G-torsor if X is nonempty and is equipped with a map (in the appropriate category) X × G → X such that

x·1 = x

x·(gh) = (x·g)·h

for all x ∈ X and all g,h ∈ G and such that the map X × G → X × X given by

is an isomorphism (of sets, or topological spaces or ..., as appropriate). Note that this means that X and G are isomorphic. However — and this is the essential point —, there is no preferred 'identity' point in X. That is, X looks exactly like G but we have forgotten which point is the identity. This concept is often used in mathematics as a way of passing to a more intrinsic point of view, under the heading 'throw away the origin'.

Since X is not a group we cannot multiply elements; we can, however, take their "quotient". That is, there is a map X × X → G which sends (x,y) to the unique element g = x \ y ∈ G such that y = x·g.

The composition of this operation with the right group action, however, yields a ternary operation X × (X × X) → X × G → X that serves as an affine generalization of group multiplication and is sufficient to both characterize a principal homogeneous space algebraically, and intrinsically characterize the group it is associated with. If is the result of this operation, then the following identities

will suffice to define a principal homogeneous space, while the additional property

identifies those spaces that are associated with abelian groups. The group may be defined as formal quotients subject to the equivalence relation

,

with the group product, identity and inverse defined, respectively, by

,

,

and the group action by