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Encyclopedia > Principal homogeneous space
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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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... This article is about the mathematical concept. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ...


An analogous definition holds in other categories where In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ...

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 × GX such that In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, a continuous function is generally defined as one for which preimages of open sets are open. ... In mathematics, a Lie group (IPA ) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... In mathematics, a regular function in the sense of algebraic geometry is an everywhere-defined, polynomial function on an algebraic variety V with values in the field K over which V is defined. ... Please refer to group theory for a general description of the topic. ...

for all xX and all g,hG and such that the map X × GX × X given by

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'.


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

Contents


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 principal homogeneous space for V acting as the additive group of translations. In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... Jump to: navigation, search A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...


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. Jump to: navigation, search A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... Jump to: navigation, search In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... This article is about the mathematical concept. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ...


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 a cartesian product. But sections will often not exist globally. For example a differential manifold M has a principal bundle of frames associated to its tangent bundle. A global section will exist, tautologically, only when M is parallelizable; which implies strong topological restrictions. In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... Section can be: A cross section (in the common sense or the physics sense) In mathematics: A conic section A section of a fiber bundle or sheaf A Caesarean section In UK law, Section 28 In the fictional Star Trek universe, Section 31 A military unit A section (land) is... In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ... In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... In mathematics, the tangent bundle of a differentiable manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... This is a glossary of terms specific to differential geometry and differential topology. ...


In number theory there is a (superficially different) reason to consider principal homogeneous spaces, for elliptic curves E defined over a field K (and more general abelian varieties). Once this was understood various other examples were collected under the heading, for other algebraic groups: quadratic forms for orthogonal groups, and Severi-Brauer varieties for projective linear groups being two. Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ... In mathematics, an elliptic curve is a plane curve defined by an equation of the form y2 = x3 + a x + b, which is non-singular; that is, its graph has no cusps or self-intersections. ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, a Severi-Brauer variety over a field K is an algebraic variety V which becomes isomorphic to projective space over an algebraic closure of K. Examples are conic sections C: provided C is non-singular, it becomes isomorphic to the projective line over any extension field L over... The projective linear group of a vector space V over a field F is the quotient group PGL(V) = GL(V)/Z(V) where GL(V) is the general linear group on V and Z(V) is the group of all nonzero scalar transformations of V. The projective special linear...


The reason of the interest for Diophantine equations, in the elliptic curve case, is that K may not be algebraically 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 a number field (the theory of the Selmer 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. In mathematics, a Diophantine equation is a polynomial equation that only allows the variables to be integers. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ... In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... In mathematics, the Weil-Châtelet group of an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. It is named for André Weil, who introduced the general group operation in it, and F. Châtelet. ...


This theory has been developed with great attention to local analysis, leading to the definition of the Tate-Shafarevich group. In general the approach of taking the torsor theory, easy over an algebraically closed field, and trying to get back 'down' to a smaller field is an aspect of descent. It leads at once to questions of Galois cohomology, since the torsors represent classes in group cohomology H1. In mathematics, the term local analysis has at least two meanings - both derived from the idea of looking at a problem relative to each prime number p first, and then later trying to integrate the information gained at each prime into a global picture. ... In mathematics, the Weil-Châtelet group of an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. It is named for André Weil, who introduced the general group operation in it, and F. Châtelet. ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i. ... In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of gluing in topology. ... In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. ... In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. ...


Other usage

The term torsor is also used without the transitivity condition, especially in sheaf theory. In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...


In that case, we talk about a (right) G-torsor E on a space X (X a scheme/manifold/topological space etc.) being a space E with a free (right, say) G action such that the map In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... Jump to: navigation, search This page is about a higher mathematics topic. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Look up Action in Wiktionary, the free dictionary Action can be used in different contexts: Action movie Action game Action Comics, an American comic book series featuring Superman Action, a British comics anthology of the 1970s In physics, the action is a crucial concept in Lagrangian mechanics In philosophy, action...

given by

is a bijection in the appropriate category. When we are in the smooth category, then a G-torsor (for G a Lie group) is then precisely a principal G bundle. Torsors in this sense correspond to classes in the cohomology H1(X,G). In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... The word category (plural categories; from Greek κατηγορια meaning assertion or accusation, hence categorical denial) has several meanings: it is used informally to mean a class of things, as in the category of all living things. See categorization. ... The word category (plural categories; from Greek κατηγορια meaning assertion or accusation, hence categorical denial) has several meanings: it is used informally to mean a class of things, as in the category of all living things. See categorization. ... In mathematics, a Lie group (IPA ) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves...


External links


  Results from FactBites:
 
Homogeneous space - definition of Homogeneous space in Encyclopedia (500 words)
In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful.
A further classical example is the space of lines in projective space of three dimensions (equivalently, the space of two-dimensionsional subspaces of a four-dimensional vector space).
It is a finite-dimensional vector space V with a group action of an algebraic group G, such that there is an orbit of G that is open for the Zariski topology (and so, dense).
Principal homogeneous space - Wikipedia, the free encyclopedia (925 words)
In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively.
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 principal homogeneous space for V acting as the additive group of translations.
The principal homogeneous space concept is a special case of that of principal bundle: it means a principal bundle with base a single point.
  More results at FactBites »

 
 

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