Group of all affine transformations of an affine space
In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line.
Over any field, the affine group may be viewed as a matrix group in a natural way. If the associated field of scalars is the real or complex field, then the affine group is a Lie group.
In mathematics, the affinegroup or general affinegroup of any affine space is the group of all invertible affine transformations from the space into...
affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space...
In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector...
on Fn in the natural manner. The affinegroup can be viewed as the group of all affine transformations of the affine space underlying the vector space...
the group G, where the number of copies is equal to the number of connected components of T. GS is affine over S if and only if G is a finite group. However...
bioinformatics Affine geometry, a geometry characterized by parallel lines Affinegroup, the group of all invertible affine transformations from any affine space...
Euclidean group is a subgroup of the group of affine transformations. It has as subgroups the translational group T(n), and the orthogonal group O(n). Any...
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent...
The affine symmetric groups are a family of mathematical structures that describe the symmetries of the number line and the regular triangular tiling...
translates of these hyperplanes. The affine Coxeter group (or affine Weyl group) is then the group generated by the (affine) reflections about all the hyperplanes...
important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly...
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance...
equivalent to affinegroup schemes. (Every affinegroup scheme over a field k is pro-algebraic in the sense that it is an inverse limit of affinegroup schemes...
the vector in Zn. The affinegroup acts transitively on the points of an affine space, and the subgroup V of the affinegroup (that is, a vector space)...
affine representation of a topological Lie group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of...
Affine algebra may refer to: Affine Lie algebra, a type of Kac–Moody algebras The Lie algebra of the affinegroup Finitely-generated algebra Affine Hecke...
Lorentz group is a 6-dimensional Lie group of linear isometries of the Minkowski space. The Poincaré group is a 10-dimensional Lie group of affine isometries...
generally affinegroup schemes) — These are the analogues of Lie groups, but over more general fields than just R or C. Although linear algebraic groups have...
cardinality of the affinegroup of an affine space with p elements. It follows that, if p is a prime number greater than 3, the symmetric group and the alternating...
mathematics, an affine braid group is a braid group associated to an affine Coxeter system. Their group rings have quotients called affine Hecke algebras...
Special affine curvature, also known as the equiaffine curvature or affine curvature, is a particular type of curvature that is defined on a plane curve...
was more restrictive than affine geometry, which in turn is more restrictive than projective geometry. Klein proposed that group theory, a branch of mathematics...
Equivalently, a linear algebraic group over k is a smooth affinegroup scheme over k. A connected linear algebraic group G {\displaystyle G} over an algebraically...
through the origin. An affine reflection group is a discrete subgroup of the affinegroup of E that is generated by a set of affine reflections of E (without...
] . {\displaystyle k[X_{1},\ldots ,X_{n}].} An affine variety or affine algebraic variety, is an affine algebraic set such that the ideal generated by...
Affine space Affine transformation AffinegroupAffine geometry Affine coordinate system Flat (geometry) Cartesian coordinate system Euclidean group Poincaré...
algebraic groups, a group element is unipotent if it acts unipotently in a certain natural group representation. A unipotent affine algebraic group is then...
Lorentz group Spinor group Symplectic group Exceptional groups G2 F4 E6 E7 E8 Affinegroup Euclidean group Poincaré group Heisenberg group Commutator...
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