Algebraic torus


In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by,, or, is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups. These groups were named by analogy with the theory of tori in Lie group theory. For example, over the complex numbers the algebraic torus is isomorphic to the group scheme, 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 spaces and buildings.

Algebraic tori over fields

In most places we suppose that the base field is perfect. This hypothesis is required to have a smooth group schemepg 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 F 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 F-rank is zero.

    Isogenies

An isogeny 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 two tori of rank 1:
  • the split torus
  • the compact form, which can be realised as the unitary group or as the special orthogonal group. It is an anisotropic torus. As a Lie group, it is also isomorphic to the 1-torus, 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 . This gives an example of isogenous, non-isomorphic tori.

Over a finite field

Over the finite field 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. The kernel of its field norm 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 lattice is the group of algebraic homomorphisms T → Gm, and the coweight lattice is the group of algebraic homomorphisms Gm → 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 nth 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 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 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:
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 is the rank of a maximal torus subgroup in ;
  • its -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. The group is called anisotropic if it contains no split tori.

Classification of semisimple groups

In the classical theory of semisimple Lie algebras over the complex field the Cartan subalgebras play a fundamental rôle in the classification via root systems and Dynkin diagrams. 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. 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 . 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
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 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 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, 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 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.