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 and
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
.
Many groups of
geometric transformation
In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often b ...
s are algebraic groups; for example,
orthogonal groups,
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
s,
projective groups,
Euclidean groups, 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 fa ...
s are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s and
Jacobian varieties.
An important class of algebraic groups is given by the
affine algebraic groups, 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
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
, 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 Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...
, which are the algebraic groups whose underlying variety is a
projective variety.
Chevalley's structure theorem
In algebraic geometry, Chevalley's structure theorem states that a smooth connected algebraic group over a perfect field has a unique normal smooth connected affine algebraic subgroup such that the quotient is an abelian variety
In mathematics, ...
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
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
), and
regular maps
(the multiplication operation) and
(the inversion operation) which satisfy the group axioms.
Examples
*The ''additive group'': the
affine line
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
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
In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the ge ...
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 differe ...
shows that inversion is regular as well on matrices with determinant 1.
*The
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of
invertible matrices
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
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
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
).
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
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group it ...
(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 In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natur ...
).
Lie algebra of an algebraic group
Similarly to the
Lie group–Lie algebra correspondence In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are ...
, 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
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have ...
over
(group schemes can more generally be defined over
commutative rings).
Yet another definition of the concept is to say that an algebraic group over
is a
group object In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is ...
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) ...
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 groups.
*
algebraic tori.
*certain
semidirect products, for instance
Jet group In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of coor ...
s, or some
solvable group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminate ...
s 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 semi ...
states that every such is (essentially) a semidirect product of a unipotent group (its
unipotent radical) with a
reductive group
In 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 is reductive if it has a representation with finite kernel which is a direct ...
. In turn reductive groups are decomposed as (again essentially) a product of their center (an algebraic torus) with a
semisimple group
In 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 is reductive if it has a representation with finite kernel which is a direc ...
. 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
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensi ...
fields, and thereby over
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
s via
local-global principle In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of eac ...
s.
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
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 ...
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
In algebraic geometry, Chevalley's structure theorem states that a smooth connected algebraic group over a perfect field has a unique normal smooth connected affine algebraic subgroup such that the quotient is an abelian variety
In mathematics, ...
asserts that every connected algebraic group is an extension of an
abelian variety by a
linear 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 ...
. More precisely, if ''K'' is a
perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds:
* Every irreducible polynomial over ''k'' has distinct roots.
* Every irreducible polynomial over ''k'' is separable.
* Every finite extension of ''k' ...
, 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
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
. 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 elem ...
). 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
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact ...
(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. 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
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
::\begin
1 & a & c\\
0 & 1 & b\\
0 & 0 & 1\\
\end
under the operation of matrix multiplication. Elements ...
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 group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
s – 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
In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name ...
, 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)=\ ...
*
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
*
Adelic algebraic group
*
Pseudo-reductive group
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