In mathematics, a linear algebraic group is a subgroup of the group of invertible n × n {\displaystyle n\times n} matrices (under matrix multiplication)...

orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally...

In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A...

In mathematics, the general linear group of degree n {\displaystyle n} is the set of n × n {\displaystyle n\times n} invertible matrices, together with...

mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group G over a perfect field...

group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted...

and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have...

Linear algebra 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...

is an outline of topics related to linear algebra, the branch of mathematics concerning linear equations and linear maps and their representations in vector...

In algebraic geometry, given a linear algebraic group G over a field k, a distribution on it is a linear functional k [ G ] → k {\displaystyle k[G]\to...

These elements are "special" in that they form an algebraic subvariety of the general linear group – they satisfy a polynomial equation (since the determinant...

linear algebraic problems like solving linear systems of equations, locating eigenvalues, or least squares optimisation. Numerical linear algebra's central...

even a linear algebraic group: see below). If G is a Lie group with Lie algebra g {\displaystyle {\mathfrak {g}}} , then the automorphism group of G has...

and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have...

abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures...

the unitary group is a linear algebraic group. The unitary group of a quadratic module is a generalisation of the linear algebraic group U just defined...

( b ) × ( a , c ) {\displaystyle (b)\times (a,c)} in the group. For a linear algebraic group G {\displaystyle G} , a Borel subgroup is defined as a subgroup...

Milne, J. S. Linear Algebraic Groups (PDF). pp. 252–253, Unipotent algebraic groups. Brion, Michel (2016-09-27). "Commutative algebraic groups up to isogeny"...

these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system...

In mathematics, the special linear Lie algebra of order n {\displaystyle n} over a field F {\displaystyle F} , denoted s l n F {\displaystyle {\mathfrak...

In linear algebra, the trace of a square matrix A, denoted tr(A), is the sum of the elements on its main diagonal, a 11 + a 22 + ⋯ + a n n {\displaystyle...

(2002). The so-called classical groups generalize the examples 1 and 2 above. They arise as linear algebraic groups, that is, as subgroups of GLn defined...

abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally...

equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components...

{\displaystyle \mathbb {C} ^{n}} . It is itself a subgroup of the general linear group, SU ⁡ ( n ) ⊂ U ⁡ ( n ) ⊂ GL ⁡ ( n , C ) . {\displaystyle \operatorname...

commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled...

of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of the general linear group GL ⁡ (...

complex semisimple Lie group is a linear algebraic group. The Lie algebra of a complex Lie group is a complex Lie algebra. A finite-dimensional vector space...

chapter of linear algebra. A group of Lie type is a group closely related to the group G(k) of rational points of a reductive linear algebraic group G with...

linear algebraic groups; Lie's theorem is the analog for linear Lie algebras. It states that if G is a connected and solvable linear algebraic group defined...