In
mathematics, an algebraic torus, where a one dimensional torus is typically denoted by
,
, or
, is a type of commutative affine
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group 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.
Ma ...
commonly found in
projective algebraic geometry and
toric geometry In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be n ...
. Higher dimensional algebraic tori can be modelled as a product of algebraic groups
. These
groups
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 ...
were named by analogy with the theory of ''tori'' in
Lie group theory (see
Cartan subgroup
In algebraic geometry, a Cartan subgroup of a connected linear algebraic group over an algebraically closed field is the centralizer of a maximal torus (which turns out to be connected). Cartan subgroups are nilpotent and are all conjugate.
Examp ...
). For example, over the complex numbers
the algebraic torus
is isomorphic to the
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 ...
, which is the scheme theoretic analogue of the Lie group
. In fact, any
-action on a complex vector space can be pulled back to a
-action from the inclusion
as real manifolds.
Tori are of fundamental importance in the theory of algebraic groups and Lie groups and in the study of the geometric objects associated to them such as
symmetric space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
s and
buildings.
Algebraic tori over fields
In most places we suppose that the base field is
perfect (for example finite or characteristic zero). This hypothesis is required to have a smooth group scheme
pg 64, since for an algebraic group
to be smooth over characteristic
, the maps
must be geometrically reduced for large enough
, meaning the image of the corresponding map on
is smooth for large enough
.
In general one has to use separable closures instead of algebraic closures.
Multiplicative group of a field
If
is a field then the ''multiplicative group'' over
is the algebraic group
such that for any field extension
the
-points are isomorphic to the group
. To define it properly as an algebraic group one can take the affine variety defined by the equation
in the affine plane over
with coordinates
. The multiplication is then given by restricting the regular rational map
defined by
and the inverse is the restriction of the regular rational map
.
Definition
Let
be a field with algebraic closure
. Then a ''
-torus'' is an algebraic group defined over
which is isomorphic over
to a finite product of copies of the multiplicative group.
In other words, if
is an
-group it is a torus if and only if
for some
. The basic terminology associated to tori is as follows.
*The integer
is called the ''rank'' or ''absolute rank'' of the torus
.
*The torus is said to be ''split'' over a field extension
if
. There is a unique minimal finite extension of
over which
is split, which is called the ''splitting field'' of
.
*The ''
-rank'' of
is the maximal rank of a split sub-torus of
. A torus is split if and only if its
-rank equals its absolute rank.
*A torus is said to be ''anisotropic'' if its
-rank is zero.
Isogenies
An
isogeny In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel.
If the groups are abelian varieties, then any morphism of the underlyi ...
between algebraic groups is a surjective morphism with finite kernel; two tori are said to be ''isogenous'' if there exists an isogeny from the first to the second. Isogenies between tori are particularly well-behaved: for any isogeny
there exists a "dual" isogeny
such that
is a power map. In particular being isogenous is an equivalence relation between tori.
Examples
Over an algebraically closed field
Over any algebraically closed field
there is up to isomorphism a unique torus of any given rank. For a rank
algebraic torus over
this is given by the group scheme
pg 230.
Over the real numbers
Over the field of real numbers
there are exactly (up to isomorphism) two tori of rank 1:
*the split torus
*the compact form, which can be realised as the
unitary group
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
or as the special
orthogonal group . It is an anisotropic torus. As a Lie group, it is also isomorphic to the 1-
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
, which explains the picture of diagonalisable algebraic groups as tori.
Any real torus is isogenous to a finite sum of those two; for example the real torus
is doubly covered by (but not isomorphic to)
. This gives an example of isogenous, non-isomorphic tori.
Over a finite field
Over the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
there are two rank-1 tori: the split one, of cardinality
, and the anisotropic one of cardinality
. The latter can be realised as the matrix group
More generally, if
is a finite field extension of degree
then the
Weil restriction from
to
of the multiplicative group of
is an
-torus of rank
and
-rank 1 (note that restriction of scalars over an inseparable field extension will yield a commutative algebraic group that is not a torus). The kernel
of its
field norm In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.
Formal definition
Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of ''K ...
is also a torus, which is anisotropic and of rank
. Any
-torus of rank one is either split or isomorphic to the kernel of the norm of a quadratic extension.
The two examples above are special cases of this: the compact real torus is the kernel of the field norm of
and the anisotropic torus over
is the kernel of the field norm of
.
Weights and coweights
Over a separably closed field, a torus ''T'' admits two primary invariants. The
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar qua ...
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
is the group of algebraic homomorphisms ''T'' → G
m, and the coweight lattice
is the group of algebraic homomorphisms G
m → ''T''. These are both free abelian groups whose rank is that of the torus, and they have a canonical nondegenerate pairing
given by
, where degree is the number ''n'' such that the composition is equal to the ''n''th power map on the multiplicative group. The functor given by taking weights is an antiequivalence of categories between tori and free abelian groups, and the coweight functor is an equivalence. In particular, maps of tori are characterized by linear transformations on weights or coweights, and the automorphism group of a torus is a general linear group over Z. The quasi-inverse of the weights functor is given by a dualization functor from free abelian groups to tori, defined by its functor of points as:
:
This equivalence can be generalized to pass between groups of multiplicative type (a distinguished class of
formal group In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one ...
s) and arbitrary abelian groups, and such a generalization can be convenient if one wants to work in a well-behaved category, since the category of tori doesn't have kernels or filtered colimits.
When a field ''K'' is not separably closed, the weight and coweight lattices of a torus over ''K'' are defined as the respective lattices over the separable closure. This induces canonical continuous actions of the absolute Galois group of ''K'' on the lattices. The weights and coweights that are fixed by this action are precisely the maps that are defined over ''K''. The functor of taking weights is an antiequivalence between the category of tori over ''K'' with algebraic homomorphisms and the category of finitely generated torsion free abelian groups with an action of the absolute Galois group of ''K''.
Given a finite separable field extension ''L''/''K'' and a torus ''T'' over ''L'', we have a
Galois module
In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...
isomorphism
:
If ''T'' is the multiplicative group, then this gives the restriction of scalars a permutation module structure. Tori whose weight lattices are permutation modules for the Galois group are called quasi-split, and all quasi-split tori are finite products of restrictions of scalars.
Tori in semisimple groups
Linear representations of tori
As seen in the examples above tori can be represented as linear groups. An alternative definition for tori is:
:''A linear algebraic group is a torus if and only if it is diagonalisable over an algebraic closure. ''
The torus is split over a field if and only if it is diagonalisable over this field.
Split rank of a semisimple group
If
is a semisimple algebraic group over a field
then:
*its ''rank'' (or ''absolute rank'') is the rank of a maximal torus subgroup in
(note that all maximal tori are conjugated over
so the rank is well-defined);
*its ''
-rank'' (sometimes called ''
-split rank'') is the maximal rank of a torus subgroup in
which is split over
.
Obviously the rank is greater than or equal the
-rank; the group is called ''split'' if and only if equality holds (that is, there is a maximal torus in
which is split over
). The group is called ''anisotropic'' if it contains no split tori (i.e. its
-rank is zero).
Classification of semisimple groups
In the classical theory of
semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra is ...
s over the complex field the
Cartan subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
s play a fundamental rôle in the classification via
root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
s and
Dynkin diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
s. This classification is equivalent to that of connected algebraic groups over the complex field, and Cartan subalgebras correspond to maximal tori in these. In fact the classification carries over to the case of an arbitrary base field under the assumption that there exists a split maximal torus (which is automatically satisfied over an algebraically closed field). Without the splitness assumption things become much more complicated and a more detailed theory has to be developed, which is still based in part on the study of adjoint actions of tori.
If
is a maximal torus in a semisimple algebraic group
then over the algebraic closure it gives rise to a root system
in the vector space
. On the other hand, if
is a maximal
-split torus its action on the
-Lie algebra of
gives rise to another root system
. The restriction map
induces a map
and the
Tits index is a way to encode the properties of this map and of the action of the Galois group of
on
. The Tits index is a "relative" version of the "absolute" Dynkin diagram associated to
; obviously, only finitely many Tits indices can correspond to a given Dynkin diagram.
Another invariant associated to the split torus
is the ''anisotropic kernel'': this is the semisimple algebraic group obtained as the derived subgroup of the centraliser of
in
(the latter is only a reductive group). As its name indicates it is an anisotropic group, and its absolute type is uniquely determined by
.
The first step towards a classification is then the following theorem
:''Two semisimple
-algebraic groups are isomorphic if and only if they have the same Tits indices and isomorphic anisotropic kernels. ''
This reduces the classification problem to anisotropic groups, and to determining which Tits indices can occur for a given Dynkin diagram. The latter problem has been solved in . The former is related to the
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 ...
groups of
. More precisely to each Tits index there is associated a unique
quasi-split group over
; then every
-group with the same index is an
inner form In mathematics, an inner form of an algebraic group G over a field K is another algebraic group H such that there exists an isomorphism \phi between G and H defined over \overline K (this means that H is a ''K-form'' of G) and in addition, for every ...
of this quasi-split group, and those are classified by the Galois cohomology of
with coefficients in the adjoint group.
Tori and geometry
Flat subspaces and rank of symmetric spaces
If
is a semisimple Lie group then its real rank is the
-rank as defined above (for any
-algebraic group whose group of real points is isomorphic to
), in other words the maximal
such that there exists an embedding
. For example, the real rank of
is equal to
, and the real rank of
is equal to
.
If
is the
symmetric space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
associated to
and
is a maximal split torus then there exists a unique orbit of
in
which is a totally geodesic flat subspace in
. It is in fact a maximal flat subspace and all maximal such are obtained as orbits of split tori in this way. Thus there is a geometric definition of the real rank, as the maximal dimension of a flat subspace in
.
Q-rank of lattices
If the Lie group
is obtained as the real points of an algebraic group
over the rational field
then the
-rank of
has also a geometric significance. To get to it one has to introduce an
arithmetic group
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm_2(\Z). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number the ...
associated to
, which roughly is the group of integer points of
, and the quotient space
, which is a Riemannian orbifold and hence a metric space. Then any
asymptotic cone
In mathematics, an ultralimit is a geometric construction that assigns to a sequence of metric spaces ''Xn'' a limiting metric space. The notion of an ultralimit captures the limiting behavior of finite configurations in the spaces ''Xn'' and use ...
of
is homeomorphic to a finite
simplicial complex with top-dimensional simplices of dimension equal to the
-rank of
. In particular,
is compact if and only if
is anisotropic.
Note that this allows to define the
-rank of any lattice in a semisimple Lie group, as the dimension of its asymptotic cone.
Buildings
If
is a semisimple group over
the maximal split tori in
correspond to the apartments of the Bruhat-Tits building
associated to
. In particular the dimension of
is equal to the
-rank of
.
Algebraic tori over an arbitrary base scheme
Definition
Given a base
scheme ''S'', an algebraic torus over ''S'' is defined to be 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 ''S'' that is
fpqc locally isomorphic to a finite product of copies of the multiplicative group scheme G
''m''/''S'' over ''S''. In other words, there exists a faithfully flat map ''X'' → ''S'' such that any point in ''X'' has a quasi-compact open neighborhood ''U'' whose image is an open affine subscheme of ''S'', such that base change to ''U'' yields a finite product of copies of ''GL''
1,''U'' = G
''m''/''U''. One particularly important case is when ''S'' is the spectrum of a field ''K'', making a torus over ''S'' an algebraic group whose extension to some finite separable extension ''L'' is a finite product of copies of G
''m''/''L''. In general, the multiplicity of this product (i.e., the dimension of the scheme) is called the
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
of the torus, and it is a locally constant function on ''S''.
Most notions defined for tori over fields carry to this more general setting.
Examples
One common example of an algebraic torus is to consider the
affine cone
In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every .
...
of a
projective scheme
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
. Then, with the origin removed, the induced projection map
gives the structure of an algebraic torus over
.
Weights
For a general base scheme ''S'', weights and coweights are defined as fpqc sheaves of free abelian groups on ''S''. These provide representations of fundamental groupoids of the base with respect the fpqc topology. If the torus is locally trivializable with respect to a weaker topology such as the etale topology, then the sheaves of groups descend to the same topologies and these representations factor through the respective quotient groupoids. In particular, an etale sheaf gives rise to a quasi-isotrivial torus, and if ''S'' is locally noetherian and normal (more generally,
geometrically unibranched), the torus is isotrivial. As a partial converse, a theorem of
Grothendieck asserts that any torus of finite type is quasi-isotrivial, i.e., split by an etale surjection.
Given a rank ''n'' torus ''T'' over ''S'', a twisted form is a torus over ''S'' for which there exists a fpqc covering of ''S'' for which their base extensions are isomorphic, i.e., it is a torus of the same rank. Isomorphism classes of twisted forms of a split torus are parametrized by nonabelian flat cohomology
, where the coefficient group forms a constant sheaf. In particular, twisted forms of a split torus ''T'' over a field ''K'' are parametrized by elements of the Galois cohomology pointed set
with trivial Galois action on the coefficients. In the one-dimensional case, the coefficients form a group of order two, and isomorphism classes of twisted forms of G
m are in natural bijection with separable quadratic extensions of ''K''.
Since taking a weight lattice is an equivalence of categories, short exact sequences of tori correspond to short exact sequences of the corresponding weight lattices. In particular, extensions of tori are classified by Ext
1 sheaves. These are naturally isomorphic to the flat cohomology groups
. Over a field, the extensions are parametrized by elements of the corresponding Galois cohomology group.
Arithmetic invariants
In his work on
Tamagawa numbers,
T. Ono introduced a type of functorial invariants of tori over finite separable extensions of a chosen field ''k''. Such an invariant is a collection of positive real-valued functions ''f''
K on isomorphism classes of tori over ''K'', as ''K'' runs over finite separable extensions of ''k'', satisfying three properties:
# Multiplicativity: Given two tori ''T''
1 and ''T''
2 over ''K'', ''f''
K(''T''
1 × ''T''
2) = ''f''
K(''T''
1) ''f''
K(''T''
2)
# Restriction: For a finite separable extension ''L''/''K'', ''f''
L evaluated on an ''L'' torus is equal to ''f''
K evaluated on its restriction of scalars to ''K''.
# Projective triviality: If ''T'' is a torus over ''K'' whose weight lattice is a projective Galois module, then ''f''
K(''T'') = 1.
T. Ono showed that the Tamagawa number of a torus over a number field is such an invariant. Furthermore, he showed that it is a quotient of two cohomological invariants, namely the order of the group
(sometimes mistakenly called the
Picard group
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
of ''T'', although it doesn't classify G
m torsors over ''T''), and the order of the
Tate–Shafarevich group.
The notion of invariant given above generalizes naturally to tori over arbitrary base schemes, with functions taking values in more general rings. While the order of the extension group is a general invariant, the other two invariants above do not seem to have interesting analogues outside the realm of fraction fields of one-dimensional domains and their completions.
See also
*
Toric geometry In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be n ...
*
Torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
*
Torus based cryptography
*
Hopf algebra
Notes
References
*A. Grothendieck, ''
SGA 3'' Exp. VIII–X
*T. Ono, ''On Tamagawa Numbers''
*T. Ono, ''On the Tamagawa number of algebraic tori'' Annals of Mathematics 78 (1) 1963.
*
* {{cite book , last=Witte-Morris , first=Dave , title=Introduction to Arithmetic Groups , publisher=Deductive Press , year=2015 , pages=492 , isbn=978-0-9865716-0-2 , url=http://deductivepress.ca/
Linear algebraic groups
Lie groups