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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.
Conversely, any affine linear transformation extends uniquely to a projective linear transformation, so the affine group is a subgroup of the projective group. For instance, Möbius transformations (transformations of the complex projective line , or Riemann sphere ) are affine (transformations of the complex plane ) if and only if they fix the ...
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
The affine group Aff(n, F) is an extension of GL(n, F) by the group of translations in F n. It can be written as a semidirect product: Aff(n, F) = GL(n, F) ⋉ F n. where GL(n, F) acts on F n in the natural manner. The affine group can be viewed as the group of all affine transformations of the affine space underlying the vector space F n.
The affine group acts transitively on the points of an affine space, and the subgroup V of the affine group (that is, a vector space) has transitive and free (that is, regular) action on these points; [14] indeed this can be used to give a definition of an affine space.
The additive group: the affine line endowed with addition and opposite as group operations is an algebraic group. It is called the additive group (because its k {\displaystyle k} -points are isomorphic as a group to the additive group of k {\displaystyle k} ), and usually denoted by G a {\displaystyle \mathrm {G} _{a}} .
If is a field then the multiplicative group over is the algebraic group such that for any field extension / the -points are isomorphic to the group . To define it properly as an algebraic group one can take the affine variety defined by the equation x y = 1 {\displaystyle xy=1} in the affine plane over F {\displaystyle F} with coordinates x , y ...
Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff(A) of the affine group of A. An example is the action of the Euclidean group E(n) on the Euclidean space E n. Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought ...
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