affine group

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  • Info In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.
  • This is a group under the operation of composition of functions, called the affine group.
  • There is a sample space of lines, one on which the affine group of the plane acts.
  • Concretely, each leaf is a local orbit of the special affine group.
  • For example, the left and right invariant Haar measures on the affine group are not equal.
  • Affine group varieties are known as linear algebraic groups, since they can be embedded as subgroups of general linear groups.
  • Nisnevich introduced his topology to provide a cohomological interpretation of the class set of an affine group scheme, which was originally defined in adelic terms.
  • The similarities group S is itself a subgroup of the affine group, so every similarity is an affine transformation.
  • Although the above equations can be used to define the Lanczos tensor, they also show that it is not unique but rather has gauge freedom under an affine group.
  • The so-called quasi-abelian functions are all known to come from extensions of abelian varieties by commutative affine group varieties.
  • In the mathematical study of transformation groups, the special affine group is the group of affine transformations of a fixed affine space which preserve volume.
  • Presuming knowledge of projectivity and the projective group of projective geometry, the affine group can be easily specified.
  • These transformations generate a subgroup, of the planar affine group, called the Lorentz group of the plane.
  • Similar to wavelets, shearlets arise from the affine group and allow a unified treatment of the continuum and digital situation leading to faithful implementations.
  • To explain accurately the relationship between affine and Euclidean geometry, we now need to pin down the group of Euclidean geometry within the affine group.
  • Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought of as a particular kind of linear representation.
  • He discovered the notion of normal subgroups and found that a solvable primitive group may be identified to a subgroup of the affine group of an affine space over a finite field of prime order.
  • The group of area-preserving transformations (whether the special linear group or the special affine group) contains such subgroups, and this opens the possibility of performing paradoxical decompositions using them.
  • There are a pair of invariants of the curve that are invariant under the full general affine group -- the group of all affine motions of the plane, not just those that are area-preserving.
  • Complete connected group schemes are in some sense opposite to affine group schemes, since the completeness implies all global sections are exactly those pulled back from the base, and in particular, they have no nontrivial maps to affine schemes.
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