The first one, successfully tackled by Hilbert in the case of a general linear group, is to prove that the algebra A is finitely generated.This is necessary if one wanted the quotient to be an affine algebraic variety.Whether a similar fact holds for arbitrary groups G was the subject of Hilbert's fourteenth problem, and Nagata demonstrated that the answer was negative in general.…

An affine algebraic group is called unipotent if all its elements are unipotent.Any unipotent algebraic group is isomorphic to a closed subgroup of the group of upper triangular matrices with diagonal entries 1, and conversely any such subgroup is unipotent.…

According to another basic theorem, any group in the category of affine varieties has a faithful finite-dimensional linear representation: we can consider it to be a matrix group over K, defined by polynomials over K and with matrix multiplication as the group operation.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.This was studied in particular by Maxwell Rosenlicht, and can be used to study ramified coverings of a curve, with abelian Galois group.…

The quotient group G/G 0 is called the group of components or component group of G. Its elements are just the connected components of G. The component group G/G 0 is a discrete group if and only if G 0 is open.If G is an affine algebraic group then G/G 0 is actually a finite group.One may similarly define the path component group as the group of path components (quotient of G by the identity path component), and in general the component group is a quotient of the path component group, but if G is locally path connected these groups agree.…

The implicit equations are the basis of algebraic geometry, whose basic subjects of study are the simultaneous solutions of several implicit equations whose left-hand sides are polynomials.These sets of simultaneous solutions are called affine algebraic sets.The solutions of differential equations generally appear expressed by an implicit function.…

Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphic representations, and the arithmetic of quadratic forms.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.

Like in the non graded case, this Hopf algebra can be described purely algebraically as the universal enveloping algebra of the Lie superalgebra.In a similar way one can define an affine algebraic supergroup as a group object in the category of super algebraic affine varieties.An affine algebraic supergroup has a similar one to one relation to its Hopf algebra of super Polynomials.…

The term quasi-unipotent means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity.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).…

In a similar way one can define an affine algebraic supergroup as a group object in the category of super algebraic affine varieties.An affine algebraic supergroup has a similar one to one relation to its Hopf algebra of super Polynomials.Using the language of schemes, which combines the geometric and algebraic point of view, algebraic supergroup schemes can be defined including super Abelian varieties.…

For any affine algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring by this ideal.Let be an algebraically closed field and let be the projective n-space over, Let in be a homogeneous polynomial of degree d. It is not well-defined to evaluate on points in in homogeneous coordinates.…

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.When one wants to work over a base ring R (commutative), there is the group scheme concept: that is, a group object in the category of schemes over R. Affine group scheme is the concept dual to a type of Hopf algebra.…

The gradient of F is then normal to the hypersurface.Similarly, an affine algebraic hypersurface may be defined by an equation, where F is a polynomial.The gradient of F is zero at a singular point of the hypersurface (this is the definition of a singular point).…

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.These groups were named by analogy with the theory of tori in Lie group theory (see maximal torus).…

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.The unipotent k-radical is the largest smooth connected unipotent normal subgroup defined over k. Over perfect fields these are the same as (connected) reductive groups, but over non-perfect fields Jacques Tits found some examples of pseudo-reductive groups that are not reductive.…

Like for affine algebraic sets, there is a bijection between the projective algebraic sets and the reduced homogeneous ideals which define them.The projective varieties are the projective algebraic sets whose defining ideal is prime.…

Now if we had a holomorphic embedding of M into C n, then the coordinate functions of C n would restrict to nonconstant holomorphic functions on M, contradicting compactness, except in the case that M is just a point.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 classification of complex manifolds is much more subtle than that of differentiable manifolds.…

The definition of the regular maps apply also to algebraic sets.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.The affine varieties is a subcategory of the category of the algebraic sets.…