In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I where M^T is the transpose of M. Euclid, detail from The School of Athens by Raphael. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... For the square matrix section, see square matrix. ... This article gives an overview of the various ways to multiply matrices. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...

The main examples of linear algebraic groups are certain of the Lie groups, where the underlying field is the real or complex field. (For example, every compact Lie group can be regarded as the group of points of a real linear algebraic group, essentially by the Peter-Weyl theorem.) These were the first algebraic groups to be extensively studied. Such groups were known for a long time before their abstract algebraic theory was developed according to the needs of major applications. They were considered by Élie Cartan, Ludwig Maurer, Wilhelm Killing, and Sophus Lie in the 1880s and 1890s in the context of differential equations and Galois theory. However, a purely algebraic theory wasn't sought for, until around 1950. The Picard-Vessiot theory did lead to algebraic groups. In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... The Peter-Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... Wilhelm Karl Joseph Killing (1847 May 10 – 1923 February 11) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry. ... Marius Sophus Lie (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician who largely created the theory of continuous symmetry, and applied it to the study of geometric structures and differential equations. ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, Galois theory is a branch of abstract algebra. ... Motivation and Basic Idea In mathematics, the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. ...

The first basic theorem of the subject is that any affine algebraic group is a linear algebraic group: that is, any affine variety V that has an algebraic group law has a faithful linear representation, over the same field. For example the additive group of an n-dimensional vector space has a faithful representation as n+1×n+1 matrices. 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. ... This article is about algebraic varieties. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ...

One can define the Lie algebra of an algebraic group purely algebraically (it consists of the dual number points based at the identity element); and this theorem shows that we get a matrix Lie algebra. A linear algebraic group G consists of a finite number of irreducible components, that are in fact also the connected components: the one G^o containing the identity will be a normal subgroup of G. In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... A variety of dualities in mathematics are listed at duality (mathematics). ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...

One of the first uses for the theory was to define the Chevalley groups. In mathematics, a group of Lie type is a finite group related to the points of a simple algebraic group with values in a finite field. ...

The deeper structure theory applies to connected linear algebraic groups G, and begins with the definition of Borel subgroups B. These turn out to be maximal as connected solvable subgroups (i.e., subgroups with composition series having as factors one-dimensional subgroups, all of which are groups of additive or multiplicative type); and also minimal such that G/B is a projective variety. In mathematics, a Borel subgroup (named after Armand Borel) of an algebraic group G is a maximal solvable subgroup. ... In mathematics, a composition series of a group G is a chain of subgroups of G satisfying where stands for normal subgroup, such that each quotient group Hi+1/Hi is a simple group. ... This article is about algebraic varieties. ...

Lie groups that aren't algebraic

There are several classes of examples of Lie groups that aren't the real or complex points of an algebraic group.

• Any Lie group with an infinite group of components G/G^o cannot be realized as an algebraic group (see identity component).
• The center of a linear algebraic group is again a linear algebraic group. Thus, any group whose center has infinitely many components is not a linear algebraic group. An interesting example is the universal cover of SL₂(R). This is a Lie group that maps infinite-to-one to SL₂(R), since the fundamental group is here infinite cyclic - and in fact the cover has no faithful matrix representation.
• The general solvable Lie group need not have a group law expressible by polynomials.

See also: In mathematics, the identity component of a topological group G is the connected component C that contains the identity element e. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ...

• Differential Galois theory

// Motivation and basic idea In mathematics, the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. ...

Reference

A good introduction to the theory of linear algebraic groups is:

• Borel, Armand. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. ISBN 0-387-97370-2.