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:''For the term "torsor" in algebraic geometry, see torsor (algebraic geometry).'' In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every 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 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) such that :''x''·1 = ''x'' :''x''·(''gh'') = (''x''·''g'')·''h'' for all and all and such that the map 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'' except that which point is the identity has been forgotten. 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 that sends to the unique element such that . The composition of this operation with the right group action, however, yields a ternary operation 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 : ==Examples== Every group ''G'' can itself be thought of as a left or right ''G''-torsor under the natural action of left or right multiplication. Another example is the affine space concept: the idea of the affine space ''A'' underlying a vector space ''V'' can be said succinctly by saying that ''A'' is a principal homogeneous space for ''V'' acting as the additive group of translations. The flags of any regular polytope form a torsor for its symmetry group. Given a vector space ''V'' we can take ''G'' to be the general linear group GL(''V''), and ''X'' to be the set of all (ordered) bases of ''V''. Then ''G'' acts on ''X'' in the way that it acts on vectors of ''V''; and it acts transitively since any basis can be transformed via ''G'' to any other. What is more, a linear transformation fixing each vector of a basis will fix all ''v'' in ''V'', hence being the neutral element of the general linear group GL(''V'') : so that ''X'' is indeed a ''principal'' homogeneous space. One way to follow basis-dependence in a linear algebra argument is to track variables ''x'' in ''X''. Similarly, the space of orthonormal bases (the Stiefel manifold of ''n''-frames) is a principal homogeneous space for the orthogonal group. In category theory, if two objects ''X'' and ''Y'' are isomorphic, then the isomorphisms between them, Iso(''X'',''Y''), form a torsor for the automorphism group of ''X'', Aut(''X''), and likewise for Aut(''Y''); a choice of isomorphism between the objects gives an isomorphism between these groups and identifies the torsor with these two groups, and giving the torsor a group structure (as it is a base point). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「principal homogeneous space」の詳細全文を読む スポンサード リンク
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