In mathematics, a linear algebraic group is a subgroup of the group of invertible matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of .
Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and Kolchin (1948). In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it exists today.
One of the first uses for the theory was to define the Chevalley groups.
and 26 Related for: Linear algebraic group information
In mathematics, a linearalgebraicgroup is a subgroup of the group of invertible n × n {\displaystyle n\times n} matrices (under matrix multiplication)...
orthogonal groups, general lineargroups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraicgroups occur naturally...
general lineargroup of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because...
mathematics, a reductive group is a type of linearalgebraicgroup over a field. One definition is that a connected linearalgebraicgroup G over a perfect field...
multiplicative group of R (that is, R excluding 0). These elements are "special" in that they form an algebraic subvariety of the general lineargroup – they...
Linearalgebra is the branch of mathematics concerning linear equations such as: a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b...
group of rational points of a reductive linearalgebraicgroup with values in a finite field. The phrase group of Lie type does not have a widely accepted...
In algebraic geometry, given a linearalgebraicgroup G over a field k, a distribution on it is a linear functional k [ G ] → k {\displaystyle k[G]\to...
and the methods of group theory have influenced many parts of algebra. Linearalgebraicgroups and Lie groups are two branches of group theory that have...
linearalgebraic problems like solving linear systems of equations, locating eigenvalues, or least squares optimisation. Numerical linearalgebra's central...
commutative affine algebraicgroup commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled...
(2002). The so-called classical groups generalize the examples 1 and 2 above. They arise as linearalgebraicgroups, that is, as subgroups of GLn defined...
is an outline of topics related to linearalgebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector...
and the methods of group theory have influenced many parts of algebra. Linearalgebraicgroups and Lie groups are two branches of group theory that have...
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...
these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linearalgebraicgroup, etc. is the Weyl group of the root system...
abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures...
schemes, and they generalize algebraicgroups, in the sense that all algebraicgroups have group scheme structure, but group schemes are not necessarily...
empirical sciences. Algebra is the branch of mathematics that studies algebraic operations and algebraic structures. An algebraic structure is a non-empty...
chapter of linearalgebra. A group of Lie type is a group closely related to the group G(k) of rational points of a reductive linearalgebraicgroup G with...
Algebras], translated by Jones, G. A., Springer, ISBN 978-3-540-67827-4. Affine Weyl group Finite Coxeter group Hasse diagram Linearalgebraicgroup Nilpotent...
complex semisimple Lie group is a linearalgebraicgroup. The Lie algebra of a complex Lie group is a complex Lie algebra. A finite-dimensional vector space...
even a linearalgebraicgroup: see below). If G is a Lie group with Lie algebra g {\displaystyle {\mathfrak {g}}} , then the automorphism group of G has...
( b ) × ( a , c ) {\displaystyle (b)\times (a,c)} in the group. For a linearalgebraicgroup G {\displaystyle G} , a Borel subgroup is defined as a subgroup...
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