- Principal homogeneous space
In
mathematics , a principal homogeneous space, or torsor, for a group "G" is a set "X" on which "G" acts freely and transitively. That is, "X" is ahomogeneous space for "G" such that the stabilizer of any point is trivial.An analogous definition holds in other categories where, for example,
*"G" is atopological group , "X" is atopological space and the action is continuous,
*"G" is aLie group , "X" is asmooth manifold and the action is smooth,
*"G" is analgebraic group , "X" is analgebraic 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 "X" is nonempty and is equipped with a map (in the appropriate category) "X" × "G" → "X" such that:xcdot 1 = x:xcdot(gh) = (xcdot g)cdot hfor all "x" ∈ "X" and all "g,h" ∈ "G" and such that the map "X" × "G" → "X" × "X" given by:x,g) mapsto (x,xcdot g)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 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" = "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 x/y cdot z is the result of this operation, then the following identities:x/y cdot y = x = y/y cdot x:v/w cdot (x/y cdot z) = (v/w cdot x)/y cdot zwill suffice to define a principal homogeneous space, while the additional property:x/y cdot z = z/y cdot xidentifies those spaces that are associated with abelian groups. The group may be defined as formal quotients x ackslash y subject to the equivalence relation:x/w cdot y) ackslash z = y ackslash (w/x cdot z),with the group product, identity and inverse defined, respectively, by:w ackslash y) cdot (x ackslash z) = y ackslash (w/x cdot z) = (x/w cdot y)ackslash z,:e = x ackslash x,:x ackslash y)^{-1} = y ackslash x,and the group action by:xcdot (y ackslash z) = x/y cdot z.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 avector space "V" can be said succinctly by saying that "A" is principal homogeneous space for "V" acting as the additive group of translations.Given a
vector space "V" we can take "G" to be thegeneral 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 alinear algebra argument is to track variables "x" in "X".Applications
The principal homogeneous space concept is a special case of that of
principal bundle : it means a principal bundle with base a single point. In other words the local theory of principal bundles is that of a family of principal homogeneous spaces depending on some parameters in the base. The 'origin' can be supplied by a section of the bundle—such sections are usually assumed to exist "locally on the base"—the bundle being "locally trivial", so that the local structure is that of acartesian product . But sections will often not exist globally. For example adifferential manifold M has a principal bundle of frames associated to itstangent bundle . A global section will exist (by definition) only when M isparallelizable , which implies strong topological restrictions.In
number theory there is a (superficially different) reason to consider principal homogeneous spaces, forelliptic curve s E defined over a field K (and more general abelian varieties). Once this was understood various other examples were collected under the heading, for otheralgebraic group s:quadratic form s fororthogonal group s, and Severi-Brauer varieties forprojective linear group s being two.The reason of the interest for
Diophantine equation s, in the elliptic curve case, is that K may not bealgebraically closed . There can exist curves C that have no point defined over K, and which become isomorphic over a larger field to E, which by definition has a point over K to serve as identity element for its addition law. That is, for this case we should distinguish C that have genus 1, from elliptic curves E that have a K-point (or, in other words, provide a Diophantine equation that has a solution in K). The curves C turn out to be torsors over E, and form a set carrying a rich structure in the case that K is anumber field (the theory of theSelmer group ). In fact a typical plane cubic curve C over Q has no particular reason to have a rational point; the standard Weierstrass model always does, namely the point at infinity, but you need a point over K to put C into that form "over" K.This theory has been developed with great attention to
local analysis , leading to the definition of theTate-Shafarevich group . In general the approach of taking the torsor theory, easy over analgebraically closed field , and trying to get back 'down' to a smaller field is an aspect of descent. It leads at once to questions ofGalois cohomology , since the torsors represent classes ingroup cohomology H1.Other usage
The concept of a principal homogeneous space can also be globalized as follows. Let "X" be a "space" (a scheme/
manifold /topological space etc.), and let "G" be a group over "X", i.e., agroup object in the category of spaces over "X".In this case, a (right, say) "G"-torsor "E" on "X" is a space "E" (of the same type) over "X" with a (right) "G" action such that the morphism:E imes_X G ightarrow E imes_X E
given by
:x,g) mapsto (x,xg)
is an
isomorphism in the appropriate category, and such that "E" is locally trivial on "X", in that "E"→ "X" acquires a section locally on "X". Torsors in this sense correspond to classes in thecohomology group "H"1("X,G").When we are in the smooth manifold category, then a "G"-torsor (for "G" a
Lie group ) is then precisely a principal "G"-bundle as defined above.ee also
*
Homogeneous space
*Heap (mathematics) External links
* [http://math.ucr.edu/home/baez/torsors.html Torsors made easy] by John Baez
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