In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties.
An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called linear algebraic groups.[1] Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem states that every algebraic group can be constructed from groups in those two families.
study of algebraicgroups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraicgroups; for example...
special orthogonal group SO(n), and the symplectic group Sp(2n). Simple algebraicgroups and (more generally) semisimple algebraicgroups are reductive. Claude...
defined by polynomials, that is, that these are algebraicgroups. The founders of the theory of algebraicgroups include Maurer, Chevalley, and Kolchin (1948)...
the unitary group is a linear algebraicgroup. The unitary group of a quadratic module is a generalisation of the linear algebraicgroup U just defined...
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known...
commutative affine algebraicgroup commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled...
are some of the main areas studied in algebraic topology: In mathematics, homotopy groups are used in algebraic topology to classify topological spaces...
universal algebra, an algebraic structure is called an algebra; this term may be ambiguous, since, in other contexts, an algebra is an algebraic structure...
equals its transpose). The orthogonal group is an algebraicgroup and a Lie group. It is compact. The orthogonal group in dimension n has two connected components...
Zhenheng Li; Benjamin Steinberg; Qiang Wang (2014). Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics. Springer. p. 142. ISBN 978-1-4939-0938-4...
In mathematics, the groupalgebra can mean either A group ring of a group over some ring. A groupalgebra of a locally compact group. This disambiguation...
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as...
quantum groups, deforming the algebra of functions on the corresponding semisimple algebraicgroup or a compact Lie group. The discovery of quantum groups was...
abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures...
the spin group is not simply connected. In this case the algebraicgroup Spinp,q is simply connected as an algebraicgroup, even though its group of real...
{\displaystyle {\mathfrak {gl}}(n,F)} can be viewed as the Lie algebra of the algebraicgroup G L ( n ) F {\displaystyle \mathrm {GL} (n)_{F}} . The Lie bracket...
Lie groups (or algebraicgroups) and Lie algebras. They are used in algebraic number theory and algebraic topology. A one-dimensional formal group law...
these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraicgroup, etc. is the Weyl group of the root system...
In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root...
In algebraicgroup theory, approximation theorems are an extension of the Chinese remainder theorem to algebraicgroups G over global fields k. Eichler...
the groupalgebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such...
group of rational points of a reductive linear algebraicgroup with values in a finite field. The phrase group of Lie type does not have a widely accepted...
prefix Lie in Lie algebra are purely algebraic. For example, an infinite-dimensional Lie algebra may or may not have a corresponding Lie group. That is, there...
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of...
( b ) × ( a , c ) {\displaystyle (b)\times (a,c)} in the group. For a linear algebraicgroup G {\displaystyle G} , a Borel subgroup is defined as a subgroup...
empirical sciences. Algebra is the branch of mathematics that studies algebraic operations and algebraic structures. An algebraic structure is a non-empty...
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraicgroup, for example S L 2 ( Z ) . {\displaystyle \mathrm...
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