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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an algebraic torus, where a one dimensional torus is typically denoted by \mathbf G_, \mathbb_m, or \mathbb, is a type of commutative affine
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...
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 ...
. Higher dimensional algebraic tori can be modelled as a product of algebraic groups \mathbf G_. These groups were named by analogy with the theory of ''tori'' in
Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...
theory (see
Cartan subgroup In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group G over a (not necessarily algebraically closed) field k is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connec ...
). For example, over the complex numbers \mathbb the algebraic torus \mathbf G_ 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 hav ...
\mathbb^* = \text(\mathbb ,t^, which is the scheme theoretic analogue of the Lie group U(1) \subset \mathbb. In fact, any \mathbf G_-action on a complex vector space can be pulled back to a U(1)-action from the inclusion U(1) \subset \mathbb^* 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 isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geome ...
s and
buildings A building or edifice is an enclosed structure with a roof, walls and windows, usually standing permanently in one place, such as a house or factory. Buildings come in a variety of sizes, shapes, and functions, and have been adapted throughout ...
.


Algebraic tori over fields

In most places we suppose that the base field is
perfect Perfect commonly refers to: * Perfection; completeness, and excellence * Perfect (grammar), a grammatical category in some languages Perfect may also refer to: Film and television * ''Perfect'' (1985 film), a romantic drama * ''Perfect'' (20 ...
(for example finite or characteristic zero). This hypothesis is required to have a smooth group schemepg 64, since for an algebraic group G to be smooth over characteristic p, the maps(\cdot)^:\mathcal(G) \to \mathcal(G) must be geometrically reduced for large enough r, meaning the image of the corresponding map on G is smooth for large enough r. In general one has to use separable closures instead of algebraic closures.


Multiplicative group of a field

If F is a field then the ''multiplicative group'' over F is the algebraic group \mathbf G_ such that for any field extension E/F the E-points are isomorphic to the group E^\times. To define it properly as an algebraic group one can take the affine variety defined by the equation xy = 1 in the affine plane over F with coordinates x, y. The multiplication is then given by restricting the regular rational map F^2 \times F^2 \to F^2 defined by ((x, y), (x',y')) \mapsto (xx', yy') and the inverse is the restriction of the regular rational map (x, y) \mapsto (y, x).


Definition

Let F be a field with algebraic closure \overline F. Then a ''F-torus'' is an algebraic group defined over F which is isomorphic over \overline F to a finite product of copies of the multiplicative group. In other words, if \mathbf T is an F-group it is a torus if and only if \mathbf T(\overline F) \cong (\overline F^\times)^r for some r \ge 1. The basic terminology associated to tori is as follows. *The integer r is called the ''rank'' or ''absolute rank'' of the torus \mathrm T. *The torus is said to be ''split'' over a field extension E/F if \mathbf T(E) \cong (E^\times)^r. There is a unique minimal finite extension of F over which \mathbf T is split, which is called the ''splitting field'' of \mathbf T. *The ''F-rank'' of \mathbf T is the maximal rank of a split sub-torus of \mathbf T. A torus is split if and only if its F-rank equals its absolute rank. *A torus is said to be ''anisotropic'' if its F-rank is zero.


Isogenies

An
isogeny In mathematics, particularly 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 \phi:\mathbf T \to \mathbf T' there exists a "dual" isogeny \psi: \mathbf T' \to \mathbf T such that \psi \circ \phi is a power map. In particular being isogenous is an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equ ...
between tori.


Examples


Over an algebraically closed field

Over any algebraically closed field k = \overline there is up to isomorphism a unique torus of any given rank. For a rank n algebraic torus over k this is given by the group scheme \mathbf_m^n = \text_k(k _1,t_1^,\ldots,t_n,t_n^pg 230.


Over the real numbers

Over the field of real numbers \mathbb R there are exactly (up to isomorphism) two tori of rank 1: *the split torus \mathbb R^\times *the compact form, which can be realised as the
unitary group Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semi ...
\mathbf U(1) or as the special
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
\mathrm(2). It is an anisotropic torus. As a Lie group, it is also isomorphic to the 1-
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
\mathbf T^1, 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 \mathbb C^\times is doubly covered by (but not isomorphic to) \mathbb R^\times \times \mathbb T^1. 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 (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
\mathbb F_q there are two rank-1 tori: the split one, of cardinality q-1, and the anisotropic one of cardinality q+1. The latter can be realised as the matrix group \left\ \subset \mathrm_2(\mathbb F_q) . More generally, if E/F is a finite field extension of degree d then the Weil restriction from E to F of the multiplicative group of E is an F-torus of rank d and F-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 N_ 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) o ...
is also a torus, which is anisotropic and of rank d-1. Any F-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 \mathbb C/\mathbb R and the anisotropic torus over \mathbb F_q is the kernel of the field norm of \mathbb F_ / \mathbb F_q.


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 a quantity associated with the gravitational force exerted on the object by other objects in its environment, although there is some variation and debate as to the exact definition. Some sta ...
lattice X^\bullet(T) is the group of algebraic homomorphisms ''T'' → Gm, and the coweight lattice X_\bullet(T) 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 X^\bullet(T) \times X_\bullet(T) \to \mathbb given by (f,g) \mapsto \deg(f \circ g), 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 In mathematics, the general linear group of degree n is the set of n\times n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again inve ...
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: :D(M)_S(X) := \mathrm(M, \mathbb_(X)). 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 o ...
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 i ...
isomorphism :X^\bullet(\mathrm_T) \cong \mathrm_^ X^\bullet(T). 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 \mathbf G is a semisimple algebraic group over a field F then: *its ''rank'' (or ''absolute rank'') is the rank of a maximal torus subgroup in \mathbf G (note that all maximal tori are conjugated over F so the rank is well-defined); *its ''F-rank'' (sometimes called ''F-split rank'') is the maximal rank of a torus subgroup in G which is split over F. Obviously the rank is greater than or equal the F-rank; the group is called ''split'' if and only if equality holds (that is, there is a maximal torus in \mathbf G which is split over F). The group is called ''anisotropic'' if it contains no split tori (i.e. its F-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 modules, direct sum of Simple Lie algebra, simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals ...
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 vector space, 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 ...
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 \mathbf T is a maximal torus in a semisimple algebraic group \mathbf G then over the algebraic closure it gives rise to a root system \Phi in the vector space V = X^*(\mathbf T) \otimes_ \mathbb R. On the other hand, if _F \mathbf T \subset \mathbf T is a maximal F-split torus its action on the F-Lie algebra of \mathbf G gives rise to another root system _F \Phi. The restriction map X^*(\mathbf T) \to X^*(_F\mathbf T) induces a map \Phi \to _F\Phi \cup\ and the Tits index is a way to encode the properties of this map and of the action of the Galois group of \overline F / F on \Phi. The Tits index is a "relative" version of the "absolute" Dynkin diagram associated to \Phi; obviously, only finitely many Tits indices can correspond to a given Dynkin diagram. Another invariant associated to the split torus _F \mathbf T is the ''anisotropic kernel'': this is the semisimple algebraic group obtained as the derived subgroup of the centraliser of _F \mathbf T in \mathbf G (the latter is only a reductive group). As its name indicates it is an anisotropic group, and its absolute type is uniquely determined by _F \Phi. The first step towards a classification is then the following theorem :''Two semisimple F-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 with a field extension ''L''/''K'' acts in a na ...
groups of F. More precisely to each Tits index there is associated a unique quasi-split group over F; then every F-group with the same index is an inner form of this quasi-split group, and those are classified by the Galois cohomology of F with coefficients in the adjoint group.


Tori and geometry


Flat subspaces and rank of symmetric spaces

If G is a semisimple Lie group then its real rank is the \mathbb R-rank as defined above (for any \mathbb R-algebraic group whose group of real points is isomorphic to G), in other words the maximal r such that there exists an embedding (\mathbb R^\times)^r \to G. For example, the real rank of \mathrm_n(\mathbb R) is equal to n-1, and the real rank of \mathrm(p,q) is equal to \min(p,q). If X is the
symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geome ...
associated to G and T \subset G is a maximal split torus then there exists a unique orbit of T in X which is a totally geodesic flat subspace in X. 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 X.


Q-rank of lattices

If the Lie group G is obtained as the real points of an algebraic group \mathbf G over the rational field \mathbb Q then the \mathbb Q-rank of \mathbf G 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 theor ...
\Gamma associated to \mathbf G, which roughly is the group of integer points of \mathbf G, and the quotient space M = \Gamma \backslash X, which is a Riemannian orbifold and hence a metric space. Then any
asymptotic cone In mathematics, an ultralimit is a geometric construction that assigns a limit metric space to a sequence of metric spaces X_n. The concept captures the limiting behavior of finite configurations in the X_n spaces employing an ultrafilter to bypass ...
of M is homeomorphic to a finite
simplicial complex In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
with top-dimensional simplices of dimension equal to the \mathbb Q-rank of \mathbf G. In particular, M is compact if and only if \mathbf G is anisotropic. Note that this allows to define the \mathbf Q-rank of any lattice in a semisimple Lie group, as the dimension of its asymptotic cone.


Buildings

If \mathbf G is a semisimple group over \mathbb Q_p the maximal split tori in \mathbf G correspond to the apartments of the Bruhat-Tits building X associated to \mathbf G. In particular the dimension of X is equal to the \mathbb Q_p-rank of \mathbf G.


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 hav ...
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 A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * 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 \text(X) \subset \mathbb^ of a
projective scheme In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, the de ...
X \subset \mathbb^n. Then, with the origin removed, the induced projection map \pi: (\text(X) - \) \to X gives the structure of an algebraic torus over X.


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 Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
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 H^1(S, GL_n(\mathbb)), 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 H^1(G_K, GL_n(\mathbb)) 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 Gm 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 Ext1 sheaves. These are naturally isomorphic to the flat cohomology groups H^1(S, \mathrm_\mathbb (X^\bullet(T_1), X^\bullet(T_2))). 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 H^1(G_k, X^\bullet(T)) \cong Ext^1(T, \mathbb_m) (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 ver ...
of ''T'', although it doesn't classify Gm 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 ...
*
Torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
* Torus based cryptography *
Hopf algebra In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover ...


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