Glossary of algebraic groups - AbsoluteAstronomy.com

There are a number of mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

notions to study and classify algebraic group

Algebraic group

In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...

s.

In the sequel, G denotes an algebraic group over a field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

notion	explanation	example	remarks
linear algebraic group Linear algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...	An algebraic group that is affine	GL_n General linear group In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible... , non-example: elliptic curve Elliptic curve In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...	Linear algebraic groups are precisely the closed subgroups of GL_n.
character group Character group In mathematics, a character group is the group of representations of a group by complex-valued functions. These functions can be thought of as one-dimensional matrix representations and so are special cases of the group characters which arises in the related context of character theory... X_∗(G)	The group of characters, i.e., group homomorphisms G → G_m	X^∗(G_m) is isomorphic to Z
diagonalizable group Diagonalizable group In mathematics, an affine group is said to be diagonalizable if it is isomorphic to a subgroup of Dn, the group of diagonal matrices. A diagonalizable group defined over k is said to split over k or k-split if the isomorphism is defined over k. This coincides with the usual notion of split for an...	A closed subgroup of D_n, the group of diagonal matrices Diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero... (of size n-by-n)
semisimple group
unipotent group
commutative	The underlying (abstract) group is abelian Abelian group In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers... .	G_a (the additive group Additive group An additive group may refer to:an abelian group, when it is written using the symbol + for its binary operationa group scheme representing the underlying-additive-group functor... ), G_m (the multiplicative group Multiplicative group In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the contextany group \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,the underlying group under multiplication of the invertible elements of... ), any complete algebraic group (see abelian variety Abelian variety In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions... )
torus Torus In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...	A group that becomes isomorphic to G_mⁿ (n-fold cartesian product) when passing to the algebraic closure Algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics.... of k.		G is said to be split by some bigger field k' , if G becomes isomorphic to G_mⁿ as an algebraic group over k'.
Lie algebra Lie algebra In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the... Lie(G)	The tangent space Tangent space In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.... of G at the unit element.	Lie (GL_n) is the space of all n-by-n matrices	Equivalently, the space of all left-invariant derivations.

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