# Linear algebraic group

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 MTM = I where MT is the transpose of M.

The main examples of linear algebraic groups are certain 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. Compact Lie groups 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 was first developed by Kolchin (1948), with Armand Borel as one of its pioneers. Picard–Vessiot theory party inspired algebraic groups (giving the archaic term "Vessiot variety" for "linear algebraic group").

One of the first uses for the theory was to define the Chevalley groups.

## Examples

The general linear group ${\displaystyle GL_{n}}$ consisting of all invertible n-by-n matrices form a linear algebraic group. It contains the subgroups

${\displaystyle U_{n}(k)\subset T_{n}(k)\subset GL_{n}(k)}$

consisting of matrices of the form

${\displaystyle \left({\begin{array}{cccc}1&*&\dots *\\0&1&\ddots &*\\\dots &\ddots &\ddots &*\\0&\dots &0&1\end{array}}\right)}$ and ${\displaystyle \left({\begin{array}{cccc}*&*&\dots *\\0&*&\ddots &*\\\dots &\ddots &\ddots &*\\0&\dots &0&*\end{array}}\right)}$.

The group ${\displaystyle \mathbb {G} _{m}=GL_{1}}$ is called the multiplicative group. The group ${\displaystyle \mathbb {G} _{a}}$ of elements of K (with the addition) can also expressed as a matrix group, namely the matrices of the form

${\displaystyle {\begin{bmatrix}1&x\\0&1\end{bmatrix}}.}$

## Definitions

A linear algebraic group is, by definition, an affine algebraic variety G which is endowed with two maps

${\displaystyle \mu :G\times G\to G,i:G\to G}$

which satisfy the usual rules of a multiplication and inverse in a group. Being affine means that the ring of functions on G is generated, as an algebra, by finitely many generators, i.e.,

${\displaystyle k[G]=k[x_{1},\dots ,x_{n}]/(f_{1},\dots ,f_{m}).}$

In more geometric terms, being affine means that G is a (Zariski) closed subset of an affine space ${\displaystyle \mathbb {A} ^{n}}$. Being a variety, G is therefore reduced. Non-reduced schemes with a group structure are referred to as group schemes.

The functor ${\displaystyle G\mapsto k[G]}$ yields an equivalence between the category of affine group schemes and the category of Hopf algebras, i.e., algebras which are additionally endowed with a compatible coalgebra structure. For example, the Hopf algebra corresponding to the multiplicative group ${\displaystyle \mathbf {G} _{m}}$ is ${\displaystyle k[X,X^{-1}]}$ with comultiplication given by

${\displaystyle X\mapsto X\otimes 1+1\otimes X.}$

The classical theory of linear algebraic groups is concerned with the case that group G the base field k is algebraically closed, which is assumed throughout, unless the contrary is explicitly stated.

### Comparison with matrix groups

The first basic theorem of the subject is that any linear algebraic group in the above sense (some authors also refer to them as affine algebraic groups) is a linear algebraic group in the more traditional sense that G has a faithful linear representation V, which is also a morphism of varieties, i.e., G is a closed subgroup of ${\displaystyle GL_{n}(k)}$. For example, the additive group of an n-dimensional vector space has a faithful representation as (n+1)×(n+1) matrices. This theorem therefore relates the more abstract modern approach based on algebraic geometry with the classical approach which describes linear algebraic groups in terms of matrices.

### Basic notions

The identity component ${\displaystyle G^{\circ }}$, i.e., the connected component containing the identity element of an algebraic group G can be shown to be a normal subgroup of finite index, so there is a group extension

${\displaystyle 1\to G^{\circ }\to G\to F\to 1,}$

where F is a finite group. Because of this, the study of algebraic groups mostly focusses on connected groups.

Various notions from abstract group theory can be applied to linear algebraic groups as well. For example, the normalizer, the center, and the centralizer of a closed subgroup H of some linear algebraic group G are again closed and therefore again linear algebraic groups. The notion of solvable groups is also transferred from abstract group theory: an algebraic group is solvable if it has a composition series having as factors one-dimensional subgroups, all of which are groups of additive or multiplicative type. The Borel fixed-point theorem states that a connected solvable group acting on a non-empty complete variety admits a fixed point.

## Borel subgroups

Connected groups are studied by looking at the Borel subgroups, i.e., the maximal connected normal solvable subgroups. For example, a Borel subgroup of ${\displaystyle GL_{n}(k)}$ is the subgroup ${\displaystyle T_{n}(k)}$ of upper triangular matrices (all entries below the diagonal are zero). This example is somewhat prototypical: much the same way as any linear algebraic group is a closed subgroup of some ${\displaystyle GL_{n}(k)}$, any connected solvable group is a subgroup of some ${\displaystyle T_{n}(k)}$, according to the Lie-Kolchin theorem. Considering a Borel subgroup B of G allows to reduce certain questions about algebraic groups to solvable groups. For example, any torus T is contained in some Borel subgroup B. Of particular importance are the maximal tori (i.e., tori not properly contained in another torus).

For a subgroup H of G, the quotient space G/H can again be endowed with the structure of an algebraic variety. It turns out that the Borel subgroups are the minimal among the subgroups such this quotient is a projective variety, i.e., closed in some ${\displaystyle \mathbb {P} _{k}^{n}}$. Subgroups of G containing a Borel subgroup are called parabolic. For example, the parabolic subgroups of ${\displaystyle GL_{3}(k)}$ containing the Borel subgroup ${\displaystyle T_{3}(k)}$ are

${\displaystyle \left\{{\begin{bmatrix}*&*&*\\0&*&*\\0&*&*\end{bmatrix}}\right\}}$ and ${\displaystyle \left\{{\begin{bmatrix}*&*&*\\*&*&*\\0&0&*\end{bmatrix}}\right\}.}$

## Unipotent groups

Yet another prototypical example is the subgroup ${\displaystyle U_{n}(k)\subset T_{n}(k)}$ whose diagonal entries all equal 1. The matrices A in this group are unipotent, i.e., ${\displaystyle A-{\text{id}}}$ is nilpotent. There is a split group extension

${\displaystyle 1\to U_{n}(k)\to T_{n}(k)\to D_{n}(k)\to 1,}$

which exhibits ${\displaystyle T_{n}(k)}$ as a semidirect product of ${\displaystyle U_{n}}$ and ${\displaystyle D_{n}}$, where ${\displaystyle D_{n}(k)}$ consists of diagonal matrices. Once again, this situation is prototypical: any connected solvable group G is a semidirect product

${\displaystyle G=T\ltimes G_{u}}$

with a maximal torus T and its subgroup ${\displaystyle G_{u}}$ of unipotent elements.

Closed subgroups of ${\displaystyle D_{n}}$ are called diagonalizable. Any group which is isomorphic to ${\displaystyle G_{m}^{n}}$, an n-fold product of the multiplicative group, is called a torus.

## Semisimple and reductive groups

The radical R(G) of G is the largest normal connected solvable subgroup of G. A related notion is the unipotent radical ${\displaystyle R_{u}(G)}$ which consists of the unipotent elements of ${\displaystyle R(G)}$. If ${\displaystyle R(G)}$ (respectively ${\displaystyle R_{u}(G)}$) is trivial, G is called semisimple (respectively, reductive). For example, the radical of ${\displaystyle GL_{n}(k)}$ equals its center (isomorphic to ${\displaystyle G_{m}}$), which contains no unipotent elements other than 1, so ${\displaystyle GL_{n}}$ is reductive. By similar reasons, ${\displaystyle SL_{n}(k)}$ is semisimple.

## The Lie algebra of G

The Lie algebra ${\displaystyle {\mathfrak {g}}}$ of an algebraic group G can be defined in several equivalent ways: as the tangent space ${\displaystyle T_{e}(G)}$ at the identity element, or as the space of left invariant derivations, i.e., derivations ${\displaystyle \partial :k[G]\to k[G]}$ of the coordinate ring of G such that

${\displaystyle \partial \lambda _{x}=\lambda _{x}\partial ,}$

where ${\displaystyle \lambda _{x}:k[G]\to k[G]}$ is induced by the left multiplication with any ${\displaystyle x\in G}$.

The passage from G to ${\displaystyle {\mathfrak {g}}}$ is thus a process of differentiation. For example, for fixed ${\displaystyle x\in G}$ the derivative of the conjugation ${\displaystyle \mu :G\to G,g\mapsto xgx^{-1}}$ is called the adjoint representation:

${\displaystyle Ad:G\to \operatorname {Aut} ({\mathfrak {g}}).}$

The connection between an algebraic group G and its Lie algebra is particularly close when the field k is of characteristic zero. For example, the closed connected subgroups H of a connected linear algebraic group G are in bijection with Lie subalgebras ${\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}}$.

## Theory over general fields

The theory of linear algebraic groups over a non-algebraically closed field k is a combination of the corresponding theory over ${\displaystyle {\overline {k}}}$ and descent theory. For example, one must distinguish between split tori (ones isomorphic to ${\displaystyle G_{m}}$) and non-split tori. The latter become split after passing to a finite extension of k. If there is no splitting maximal torus, one studies the splitting tori and the maximal ones of them. If there is a rank at least 1 split torus in the group, the group is called isotropic and anisotropic if this is not the case. Any anisotropic or isotropic linear algebraic group over a field becomes split over the algebraic closure, so this distinction is interesting from the point of view of Algebraic number theory.

## Group actions

Let G be a unipotent group acting on an affine variety. Then every G-orbit in the variety is closed.

## Non-algebraic Lie groups

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/Go 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.