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# Principal homogeneous space

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. [1]

62 relations: Abelian variety, Affine space, Algebraic group, Algebraic number field, Algebraic variety, Algebraically closed field, American Mathematical Society, Automorphism, Basis (linear algebra), Cambridge University Press, Cartesian product, Category (mathematics), Category theory, Classifying space, Cohomology, Continuous function, Descent (mathematics), Differentiable manifold, Diophantine equation, Elliptic curve, Flag (geometry), Frame bundle, Galois cohomology, General linear group, Genus (mathematics), Group (mathematics), Group action, Group cohomology, Group object, Heap (mathematics), Homogeneous space, Identity (mathematics), Isomorphism, K-frame, Lie group, Linear algebra, Local analysis, Manifold, Mathematics, Morphism of algebraic varieties, Non-abelian group, Number theory, Orthogonal group, Orthonormal basis, Parallelizable manifold, Principal bundle, Projective linear group, Quadratic form, Rational point, Regular polytope, ... Expand index (12 more) »

## Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions.

## Affine space

In mathematics, an affine space is a geometric structure that generalizes the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

## Algebraic group

In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.

## Algebraic number field

In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

## Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry.

## Algebraically closed field

In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.

## American Mathematical Society

The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.

## Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

## Basis (linear algebra)

In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.

## Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

## Cartesian product

In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.

## Category (mathematics)

In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is an algebraic structure similar to a group but without requiring inverse or closure properties.

## Category theory

Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).

## Classifying space

In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e. a topological space all of whose homotopy groups are trivial) by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG → BG.

## Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.

## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

## Descent (mathematics)

In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology.

## Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

## Diophantine equation

In mathematics, a Diophantine equation is a polynomial equation, usually in two or more unknowns, such that only the integer solutions are sought or studied (an integer solution is a solution such that all the unknowns take integer values).

## Elliptic curve

In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections.

## Flag (geometry)

In (polyhedral) geometry, a flag is a sequence of faces of a polytope, each contained in the next, with exactly one face from each dimension.

## Frame bundle

In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex.

## Galois cohomology

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.

## General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.

## Genus (mathematics)

In mathematics, genus (plural genera) has a few different, but closely related, meanings.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

## Group cohomology

In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology.

## Group object

In category theory, a branch of mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets.

## Heap (mathematics)

In abstract algebra, a heap (sometimes also called a groud) is a mathematical generalization of a group.

## Homogeneous space

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively.

## Identity (mathematics)

In mathematics an identity is an equality relation A.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## K-frame

In linear algebra, a branch of mathematics, a k-frame is an ordered set of k linearly independent vectors in a space; thus k &le; n, where n is the dimension of the vector space, and if k.

## Lie group

In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.

## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

## Local analysis

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.

## Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Morphism of algebraic varieties

In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials.

## Non-abelian group

In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a.

## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

## Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

## Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.

## Parallelizable manifold

In mathematics, a differentiable manifold M of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at any point p of M the tangent vectors provide a basis of the tangent space at p. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a section on M. A particular choice of such a basis of vector fields on M is called a parallelization (or an absolute parallelism) of M.

## Principal bundle

In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group.

## Projective linear group

In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V).

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.

## Rational point

In number theory and algebraic geometry, a rational point of an algebraic variety is a solution of a set of polynomial equations in a given field.

## Regular polytope

In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.

## Scheme (mathematics)

In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.

## Selmer group

In arithmetic geometry, the Selmer group, named in honor of the work of by, is a group constructed from an isogeny of abelian varieties.

## Severi–Brauer variety

In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central simple algebras in such a way that the algebra splits over K if and only if the variety has a point rational over K.Jacobson (1996) p.113 studied these varieties, and they are also named after Richard Brauer because of their close relation to the Brauer group.

## Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

## Stiefel manifold

In mathematics, the Stiefel manifold Vk(Rn) is the set of all orthonormal ''k''-frames in Rn.

## Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M. As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.

## Tate–Shafarevich group

In arithmetic geometry, the Tate–Shafarevich group Ш(A/K), introduced by and, of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group WC(A/K).

## Ternary operation

In mathematics, a ternary operation is an ''n''-ary operation with n.

## Topological group

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology.

## Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.

## Torsor (algebraic geometry)

In algebraic geometry, given a smooth algebraic group G, a G-torsor or a principal G-bundle P over a scheme X is a scheme (or even algebraic space) with an action of G that is locally trivial in the given Grothendieck topology in the sense that the base change Y \times_X P along "some" covering map Y \to X is the trivial torsor Y \times G \to Y (G acts only on the second factor).

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## References

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