In
mathematics, an algebraic group is an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
endowed with a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
and
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
.
Many groups of
geometric transformations are algebraic groups; for example,
orthogonal groups,
general linear groups,
projective groups,
Euclidean group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations) ...
s, etc. Many
matrix group In mathematics, a matrix group is a group ''G'' consisting of invertible matrices over a specified field ''K'', with the operation of matrix multiplication. A linear group is a group that is isomorphic to a matrix group (that is, admitting a fait ...
s are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as
elliptic curves and
Jacobian varieties
In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian var ...
.
An important class of algebraic groups is given by the
affine algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n w ...
s, those whose underlying algebraic variety is an
affine variety
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. ...
; they are exactly the algebraic subgroups of the
general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the
abelian varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functio ...
, which are the algebraic groups whose underlying variety is a
projective variety.
Chevalley's structure theorem states that every algebraic group can be constructed from groups in those two families.
Definitions
Formally, an algebraic group over a field
is an algebraic variety
over
, together with a distinguished element
(the
neutral element), and
regular maps
(the multiplication operation) and
(the inversion operation) which satisfy the group axioms.
Examples
*The ''additive group'': the
affine line endowed with addition and opposite as group operations is an algebraic group. It is called the additive group (because its
-points are isomorphic as a group to the additive group of
), and usually denoted by
.
*The ''multiplicative group'': Let
be the affine variety defined by the equation
in the affine plane
. The functions
and
are regular on
, and they satisfy the group axioms (with neutral element
). The algebraic group
is called multiplicative group, because its
-points are isomorphic to the multiplicative group of the field
(an isomorphism is given by
; note that the subset of invertible elements does not define an algebraic subvariety in
).
*The
special linear group is an algebraic group: it is given by the algebraic equation
in the affine space
(identified with the space of
-by-
matrices), multiplication of matrices is regular and the formula for the inverse in terms of the
adjugate matrix
In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter today normally refers to a differ ...
shows that inversion is regular as well on matrices with determinant 1.
*The
general linear group of
invertible matrices over a field
is an algebraic group. It can be realised as a subvariety in
in much the same way as the multiplicative group in the previous example.
* A non-singular
cubic curve
In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation
:
applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an eq ...
in the projective plane
can be endowed with a geometrically defined group law which makes it into an algebraic group (see
elliptic curve).
Related definitions
An algebraic subgroup of an algebraic group
is a
subvariety of
which is also a subgroup of
(that is, the maps
and
defining the group structure map
and
, respectively, into
).
A ''morphism'' between two algebraic groups
is a regular map
which is also a group morphism. Its kernel is an algebraic sugroup of
, its image is an algebraic subgroup of
.
Quotients in the category of algebraic groups are more delicate to deal with. An algebraic subgroup is said to be normal if it is stable under every
inner automorphism (which are regular maps). If
is a normal algebraic subgroup of
then there exists an algebraic group
and a surjective morphism
such that
is the kernel of
. Note that if the field
is not algebraically closed, the morphism of groups
may not be surjective (the default of surjectivity is measured by
Galois cohomology).
Lie algebra of an algebraic group
Similarly to the
Lie group–Lie algebra correspondence, to an algebraic group over a field
is associated a
Lie algebra over
. As a vector space the Lie algebra is isomorphic to the tangent space at the identity element. The Lie bracket can be constructed from its interpretation as a space of derivations.
Alternative definitions
A more sophisticated definition of an algebraic group over a field
is that it is that of a
group scheme over
(group schemes can more generally be defined over
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
s).
Yet another definition of the concept is to say that an algebraic group over
is a
group object in the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
*C ...
of algebraic varieties over
.
Affine algebraic groups
An algebraic group is said to be affine if its underlying algebraic variety is an affine variety. Among the examples above the additive, multiplicative groups and the general and special linear groups are affine. Using the action of an affine algebraic group on its
coordinate ring
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. ...
it can be shown that every affine algebraic group is a linear (or matrix group), meaning that it is isomorphic to an algebraic subgroup of the general linear group.
For example the additive group can be embedded in
by the morphism
.
There are many examples of such groups beyond those given previously:
*orthogonal and symplectic groups are affine algebraic groups.
*
unipotent group
In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''.
In particular, a square matrix ''M'' is a unipote ...
s.
*
algebraic tori.
*certain
semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product is a particular way in ...
s, for instance
Jet groups, or some
solvable groups such as that of invertible
triangular matrices.
Linear algebraic groups can be classified to a certain extent.
Levi's theorem
In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a semis ...
states that every such is (essentially) a semidirect product of a unipotent group (its
unipotent radical) with a
reductive group. In turn reductive groups are decomposed as (again essentially) a product of their center (an algebraic torus) with a
semisimple group. The latter are classified over algebraically closed fields via their
Lie algebra. The classification over arbitrary fields is more involved but still well-understood. If can be made very explicit in some cases, for example over the real or
p-adic fields, and thereby over
number fields via
local-global principles.
Abelian varieties
Abelian varieties are connected projective algebraic groups, for instance elliptic curves. They are always commutative. They arise naturally in various situations in algebraic geometry and number theory, for example as the
Jacobian variety of a curve.
Structure theorem for general algebraic groups
Not all algebraic groups are linear groups or abelian varieties, for instance some group schemes occurring naturally in arithmetic geometry are neither.
Chevalley's structure theorem asserts that every connected algebraic group is an extension of an
abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functio ...
by a
linear algebraic group. More precisely, if ''K'' is a
perfect field, and ''G'' a connected algebraic group over ''K'', there exists a unique normal closed subgroup ''H'' in ''G'', such that ''H'' is a connected linear algebraic group and ''G''/''H'' an abelian variety.
Connectedness
As an algebraic variety
carries a
Zariski topology. It is not in general a
group topology, i.e. the group operations may not be continuous for this topology (because Zariski topology on the product is not the product of Zariski topologies on the factors).
An algebraic group is said to be ''connected'' if the underlying algebraic variety is connected for the Zariski topology. For an algebraic group this means that it is not the union of two proper algebraic subgroups.
Examples of groups which are not connected are given by the algebraic subgroup of
th roots of unity in the multiplicative group
(each point is a Zariski-closed subset so it is not connected for
). This group is generally denoted by
. Another non-connected group are orthogonal group in even dimension (the determinant gives a surjective morphism to
).
More generally every finite group is an algebraic group (it can be realised as a finite, hence Zariski-closed, subgroup of some
by
Cayley's theorem
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group.
More specifically, is isomorphic to a subgroup of the symmetric group \operatorname(G) whose element ...
). In addition it is both affine and projective. Thus, in particular for classification purposes, it is natural to restrict statements to connected algebraic group.
Algebraic groups over local fields and Lie groups
If the field
is a
local field (for instance the real or complex numbers, or a p-adic field) and
is a
-group then the group
is endowed with the analytic topology coming from any embedding into a projective space
as a quasi-projective variety. This is a group topology, and it makes
into a topological group. Such groups are important examples in the general theory of topological groups.
If
or
then this makes
into a
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
. Not all Lie groups can be obtained via this procedure, for example the universal cover of
SL2(R), or the quotient of the
Heisenberg group by a infinite normal discrete subgroup.
An algebraic group over the real or complex numbers may have closed subgroups (in the analytic topology) which do not have the same connected component of the identity as any algebraic subgroup.
Coxeter groups and algebraic groups
There are a number of analogous results between algebraic groups and
Coxeter groups – for instance, the number of elements of the symmetric group is
, and the number of elements of the general linear group over a finite field is the
''q''-factorial ; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the
field with one element, which considers Coxeter groups to be simple algebraic groups over the field with one element.
See also
*
Character variety
In the mathematics of moduli theory, given an algebraic, reductive, Lie group G and a finitely generated group \pi, the G-''character variety of'' \pi is a space of equivalence classes of group homomorphisms from \pi to G:
:\mathfrak(\pi,G)=\op ...
*
Borel subgroup
In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
*
Tame group
*
Morley rank
*
Cherlin–Zilber conjecture
In model theory, a stable group is a group that is stable in the sense of stability theory.
An important class of examples is provided by groups of finite Morley rank (see below).
Examples
*A group of finite Morley rank is an abstract group ''G ...
*
Adelic algebraic group
*
Pseudo-reductive group In mathematics, a pseudo-reductive group over a field ''k'' (sometimes called a ''k''-reductive group) is a smooth connected affine algebraic group defined over ''k'' whose ''k''-unipotent radical (i.e., largest smooth connected unipotent normal ''k ...
References
*
*
*
*
* Milne, J. S.,
Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups'
*
*
*
* {{Citation , last1=Weil , first1=André , author1-link=André Weil , title=Courbes algébriques et variétés abéliennes , publisher=Hermann , location=Paris , oclc=322901 , year=1971
Further reading
Algebraic groups and their Lie algebrasby Daniel Miller
Properties of groups