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 M^TM = I where M^T 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 $GL_{n}$ consisting of all invertible n-by-n matrices form a linear algebraic group. It contains the subgroups

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

consisting of matrices of the form

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

and

\left({\begin{array}{cccc}*&*&\dots *\\0&*&\ddots &*\\\dots &\ddots &\ddots &*\\0&\dots &0&*\end{array}}\right)

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

{\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

\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.,

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 $\mathbb A^n$ . Being a variety, G is therefore reduced. Non-reduced schemes with a group structure are referred to as group schemes.

The functor $G\mapsto k[G]$ 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 $\mathbf {G} _{m}$ is $k[X,X^{-1}]$ with comultiplication given by

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 $GL_{n}(k)$ . 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 $G^{\circ }$ , 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

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 G acting on a non-empty complete variety X admits a point x which is fixed by all $g\in G$ .

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 $GL_{n}(k)$ is the subgroup $T_{n}(k)$ 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 $GL_{n}(k)$ , any connected solvable group is a subgroup of some $T_{n}(k)$ , 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 $\mathbb {P} _{k}^{n}$ . Subgroups of G containing a Borel subgroup are called parabolic. For example, the parabolic subgroups of $GL_{3}(k)$ containing the Borel subgroup $T_{3}(k)$ are

\left\{{\begin{bmatrix}*&*&*\\0&*&*\\0&*&*\end{bmatrix}}\right\}

and

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

Diagonalizable groups

Any group which is isomorphic to $\mathbf {G} _{m}^{n}$ , an n-fold product of the multiplicative group, is called a torus. In particular, the group $D_{n}$ of diagonal matrices is a torus. Algebraic groups which are isomorphic to a closed subgroup of $D_{n}$ are called diagonalizable. Such groups are goverened^[1] by their characters, i.e., homomorphisms (of algebraic groups)

\chi :G\to \mathbf {G} _{m}.

For any (not necessarily diagonalizable) G, the group $Y(G):=\operatorname {Hom} (\mathbf {G} _{m},G)$ of cocharacters is dual to the group $X(G)$ of characters. In connection with the classification of reductive groups by root systems, this duality can be used to define the Langlands dual $G^{\vee }$ of some reductive group G.

Unipotent groups

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

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

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

G=T\ltimes G_{u}

with a maximal torus T and its subgroup $G_u$ of unipotent elements.

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

The Lie algebra of G

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

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

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

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

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 ${\mathfrak {h}}\subset {\mathfrak {g}}$ .

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 ${\overline {k}}$ and descent theory. For example, one must distinguish between split tori (ones isomorphic to $G_{m}$ ) 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.

Applications

Group actions and geometric invariant theory

In various situations in algebraic geometry, linear algebraic groups act on other algebraic varieties in the guise of a morphism

G\times X\to X.

Such actions tend to arise when G is some kind of automorphism group; for example $GL_{n}$ consists of automorphisms of an n-dimensional vector space, whereas the orthogonal group, say, consists of automorphisms preserving a scalar product.

If G is unipotent and X is affine, then every G-orbit $Gx$ is closed.

The objective of geometric invariant theory is the study of the quotient

G\backslash X

of the G-action on X. The existence of such a quotient (as an algebraic variety) is a subtle question since the ring (in case X is affine)

(k[X])^{G}

of G-invariant functions on X need not in general be finitely generated. Haboush's theorem asserts that this ring is indeed finitely generated if G is redcutive; for example this applies to $G=GL_{n}$ , in which case the result goes back to Hilbert.

Related notions

Linear algebraic groups admit variants in different directions. Dropping the existence of the inverse map $i:G\to G$ , one obtains the notion of a linear algebraic monoid.^[2]

Lie groups

A linear algebraic group G over the fields of real or complex numbers gives rise to a Lie group, essentially since (real or complex) polynomials, which are the equations defining G (including its multiplication and inverse map) are also (real or complex) differentiable functions. In fact, many notions in the algebraic theory of algebraic groups have close analogues for Lie groups.

Conversely, 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.

General algebraic groups

Dropping the assumption that G be an affine scheme leads to a very different theory. A very rich theory, both concerning the geometry (i.e., the case k algebraically closed) and the arithmetic (such as k being a finite field, number field, or local field), has been developed for algebraic groups which are also projective varieties. In marked contrast to affine algebraic groups, such projective algebraic groups are necessarily abelian, and are referred to as abelian varieties.

Tannakian categories

The category of representations of an algebraic group G, Rep_G, together with the tensor product of representations, forms a tannakian category. In fact, tannakian categories are equivalent to pro-algebraic group schemes (i.e., pro-objects in algebraic group schemes). For example, the Mumford-Tate group and the motivic Galois group are constructed using this formalism. Certain properties of a (pro-)algebraic group G can be read off its category of representations: for example, over a field of characteristic zero, Rep_G is a semi-simple category if and only if G is pro-reductive.^[3]

References

↑ In fact, the set $X(G)$ of such characters is an (abstract) abelian group for any G. The assignment
$G\mapsto X(G)$
yields an equivalence between the category of diagonalizable groups and abelian groups without p-torsion, where p is the characteristic of the ground field k.
↑ Renner, Lex (2006), Linear Algebraic Monoids, Springer .
↑ Deligne, Pierre; Milne, J. S. (1982), Tannakian categories (in Hodge Cycles, Motives, and Shimura Varieties by Pierre Deligne, James S. Milne, Arthur Ogus, Kuang-yen Shih), Lecture Notes in Math., 900, Springer-Verlag . Proposition II.2.23.

Borel, Armand, Linear Algebraic Groups (2nd ed.), New York: Springer-Verlag, ISBN 0-387-97370-2
Conrad, Brian, Reductive group schemes (PDF) , develops the theory from the scheme-theoretic viewpoint.
Humphreys, James E. (1995), Linear algebraic groups, Springer, ISBN 0-387-90108-6
Kolchin, E. R. (1948), "Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations", Annals of Mathematics. Second Series, 49: 1–42, ISSN 0003-486X, JSTOR 1969111, MR 0024884
Springer, Tony A. (1998), Linear algebraic groups (2nd ed.), New York: Birkhäuser

External links

Hazewinkel, Michiel, ed. (2001), "Linear algebraic group", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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