affine algebraic

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  • This is necessary if one wanted the quotient to be an affine algebraic variety.
  • An affine algebraic group is called unipotent if all its elements are unipotent.
  • For that reason a concept of affine algebraic group is redundant over a field -- we may as well use a very concrete definition.
  • One kind is an algebraic group, typically an extension of an abelian variety by an affine algebraic group.
  • If G is an affine algebraic group then G/G 0 is actually a finite group.
  • These sets of simultaneous solutions are called affine algebraic sets.
  • In case G is a linear algebraic group, it is an affine algebraic variety in affine N-space.
  • Affine varieties can be given a natural topology by declaring the closed sets to be precisely the affine algebraic sets.
  • In a similar way one can define an affine algebraic supergroup as a group object in the category of super algebraic affine varieties.
  • A unipotent affine algebraic group is one all of whose elements are unipotent (see below for the definition of an element being unipotent in such a group).
  • An affine algebraic supergroup has a similar one to one relation to its Hopf algebra of super Polynomials.
  • For any affine algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring by this ideal.
  • A more obvious difference between the two concepts arises because the identity component of an affine algebraic group G is necessarily of finite index in G.
  • Similarly, an affine algebraic hypersurface may be defined by an equation, where F is a polynomial.
  • In mathematics, the Hochschild-Mostow group, introduced by, is the universal pro-affine algebraic group generated by a group.
  • In mathematics, an algebraic torus is a type of commutative affine algebraic group.
  • In mathematics, a pseudo-reductive group or k-reductive group over a field k is a smooth connected affine algebraic group defined over k whose unipotent k-radical is trivial.
  • Like for affine algebraic sets, there is a bijection between the projective algebraic sets and the reduced homogeneous ideals which define them.
  • Complex manifolds that can be embedded in C n are called Stein manifolds and form a very special class of manifolds including, for example, smooth complex affine algebraic varieties.
  • The regular maps are also called morphisms, as they make the collection of all affine algebraic sets into a category, where the objects are the affine algebraic sets and the morphisms are the regular maps.
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