Principal homogeneous space

In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. That is, X is a homogeneous space for G such that the stabilizer of any point is trivial.

Related Topics:
Mathematics - Group - Acts - Homogeneous space

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An analogous definition holds in other categories where

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• 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. For concreteness, we will use right actions. To state the definition more explicitly, X is a G-torsor if there is a map (in the appropriate category) X × G → X such that

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:xcdot 1 = x

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:xcdot(gh) = (xcdot g)cdot h

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

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:(x,g) mapsto (x,xcdot g)

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is an isomorphism. Note that this means 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'.

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

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Principal homogeneous space: Examples ►