Algebraic torus

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

Let ${\displaystyle F}$ be a field with algebraic closure ${\displaystyle {\overline {F}}}$. Then a ${\displaystyle F}$-torus is an algebraic group defined over ${\displaystyle F}$ which is isomorphic over ${\displaystyle {\overline {F}}}$ to a finite product of copies of the multiplicative group.

In other words, if ${\displaystyle \mathbf {T} }$ is an ${\displaystyle F}$-group it is a torus if and only if ${\displaystyle \mathbf {T} ({\overline {F}})\cong ({\overline {F}}^{\times })^{r}}$ for some ${\displaystyle r\geq 1}$. The basic terminology associated to tori is as follows.

• The integer ${\displaystyle r}$ is called the rank or absolute rank of the torus ${\displaystyle \mathrm {T} }$.
• The torus is said to be split over a field extension ${\displaystyle E/F}$ if ${\displaystyle \mathbf {T} (E)\cong (E^{\times })^{r}}$. There is a unique minimal finite extension of ${\displaystyle F}$ over which ${\displaystyle \mathbf {T} }$ is split, which is called the splitting field of ${\displaystyle \mathbf {T} }$.
• The ${\displaystyle F}$-rank of ${\displaystyle \mathbf {T} }$ is the maximal rank of a split sub-torus of ${\displaystyle \mathbf {T} }$. A torus is split if and only if its ${\displaystyle F}$-rank equals its absolute rank.
• A torus is said to be anisotropic if its ${\displaystyle F}$-rank is zero.

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 ${\displaystyle \phi$ :\mathbf {T} \to \mathbf {T} '} there exists a "dual" isogeny ${\displaystyle \psi$ :\mathbf {T} '\to \mathbf {T} } such that ${\displaystyle \psi \circ \phi }$ is a power map. In particular being isogenous is an equivalence relation between tori.

Over any algebraically closed field ${\displaystyle k={\overline {k}}}$ there is up to isomorphism a unique torus of any given rank. For a rank ${\displaystyle n}$ algebraic torus over ${\displaystyle k}$ this is given by the group scheme ${\displaystyle \mathbf {G} _{m}={\text{Spec}}_{k}(k[t_{1},t_{1}^{-1},\ldots ,t_{n},t_{n}^{-1}])}$[1]^{pg 230}.

Over the field of real numbers ${\displaystyle \mathbb {R} }$ there are exactly (up to isomorphism) two tori of rank 1:

• the split torus ${\displaystyle \mathbb {R} ^{\times }}$
• the compact form, which can be realised as the unitary group ${\displaystyle \mathbf {U} (1)}$ or as the special orthogonal group ${\displaystyle \mathrm {SO} (2)}$. It is an anisotropic torus. As a Lie group, it is also isomorphic to the 1-torus ${\displaystyle \mathbf {T} ^{1}}$, 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 ${\displaystyle \mathbb {C} ^{\times }}$ is doubly covered by (but not isomorphic to) ${\displaystyle \mathbb {R} ^{\times }\times \mathbb {T} ^{1}}$. This gives an example of isogenous, non-isomorphic tori.

Over the finite field ${\displaystyle \mathbb {F} _{q}}$ there are two rank-1 tori: the split one, of cardinality ${\displaystyle q-1}$, and the anisotropic one of cardinality ${\displaystyle q+1}$. The latter can be realised as the matrix group

$${\displaystyle \left\{{\begin{pmatrix}t&du\\u&t\end{pmatrix}}:t,u\in \mathbb {F} _{q},t^{2}-du^{2}=1\right\}\subset \mathrm {SL} _{2}(\mathbb {F} _{q}).}$$

More generally, if ${\displaystyle E/F}$ is a finite field extension of degree ${\displaystyle d}$ then the Weil restriction from ${\displaystyle E}$ to ${\displaystyle F}$ of the multiplicative group of ${\displaystyle E}$ is an ${\displaystyle F}$-torus of rank ${\displaystyle d}$ and ${\displaystyle F}$-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 ${\displaystyle N_{E/F}}$ of its field norm is also a torus, which is anisotropic and of rank ${\displaystyle d-1}$. Any ${\displaystyle F}$-torus of rank one is either split or isomorphic to the kernel of the norm of a quadratic extension.[2] The two examples above are special cases of this: the compact real torus is the kernel of the field norm of ${\displaystyle \mathbb {C} /\mathbb {R} }$ and the anisotropic torus over ${\displaystyle \mathbb {F} _{q}}$ is the kernel of the field norm of ${\displaystyle \mathbb {F} _{q^{2}}/\mathbb {F} _{q}}$.

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.

If ${\displaystyle \mathbf {G} }$ is a semisimple algebraic group over a field ${\displaystyle F}$ then:

• its rank (or absolute rank) is the rank of a maximal torus subgroup in ${\displaystyle \mathbf {G} }$ (note that all maximal tori are conjugated over ${\displaystyle F}$ so the rank is well-defined);
• its ${\displaystyle F}$-rank (sometimes called ${\displaystyle F}$-split rank) is the maximal rank of a torus subgroup in ${\displaystyle G}$ which is split over ${\displaystyle F}$.

Obviously the rank is greater than or equal the ${\displaystyle F}$-rank; the group is called split if and only if equality holds (that is, there is a maximal torus in ${\displaystyle \mathbf {G} }$ which is split over ${\displaystyle F}$). The group is called anisotropic if it contains no split tori (i.e. its ${\displaystyle F}$-rank is zero).

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 (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 ${\displaystyle \mathbf {T} }$ is a maximal torus in a semisimple algebraic group ${\displaystyle \mathbf {G} }$ then over the algebraic closure it gives rise to a root system ${\displaystyle \Phi }$ in the vector space ${\displaystyle V=X^{*}(\mathbf {T} )\otimes _{\mathbb {Z} }\mathbb {R} }$. On the other hand, if ${\displaystyle {}_{F}\mathbf {T} \subset \mathbf {T} }$ is a maximal ${\displaystyle F}$-split torus its action on the ${\displaystyle F}$-Lie algebra of ${\displaystyle \mathbf {G} }$ gives rise to another root system ${\displaystyle {}_{F}\Phi }$. The restriction map ${\displaystyle X^{*}(\mathbf {T} )\to X^{*}(_{F}\mathbf {T} )}$ induces a map ${\displaystyle \Phi \to {}_{F}\Phi \cup \{0\}}$ and the Tits index is a way to encode the properties of this map and of the action of the Galois group of ${\displaystyle {\overline {F}}/F}$ on ${\displaystyle \Phi }$. The Tits index is a "relative" version of the "absolute" Dynkin diagram associated to ${\displaystyle \Phi }$; obviously, only finitely many Tits indices can correspond to a given Dynkin diagram.

Another invariant associated to the split torus ${\displaystyle {}_{F}\mathbf {T} }$ is the anisotropic kernel: this is the semisimple algebraic group obtained as the derived subgroup of the centraliser of ${\displaystyle {}_{F}\mathbf {T} }$ in ${\displaystyle \mathbf {G} }$ (the latter is only a reductive group). As its name indicates it is an anisotropic group, and its absolute type is uniquely determined by ${\displaystyle {}_{F}\Phi }$.

The first step towards a classification is then the following theorem[3]

Two semisimple ${\displaystyle F}$-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 Tits (1966). The former is related to the Galois cohomology groups of ${\displaystyle F}$. More precisely to each Tits index there is associated a unique quasi-split group over ${\displaystyle F}$; then every ${\displaystyle F}$-group with the same index is an inner form of this quasi-split group, and those are classified by the Galois cohomology of ${\displaystyle F}$ with coefficients in the adjoint group.

If ${\displaystyle G}$ is a semisimple Lie group then its real rank is the ${\displaystyle \mathbb {R} }$-rank as defined above (for any ${\displaystyle \mathbb {R} }$-algebraic group whose group of real points is isomorphic to ${\displaystyle G}$), in other words the maximal ${\displaystyle r}$ such that there exists an embedding ${\displaystyle (\mathbb {R} ^{\times })^{r}\to G}$. For example, the real rank of ${\displaystyle \mathrm {SL} _{n}(\mathbb {R} )}$ is equal to ${\displaystyle n-1}$, and the real rank of ${\displaystyle \mathrm {SO} (p,q)}$ is equal to ${\displaystyle \min(p,q)}$.

If ${\displaystyle X}$ is the symmetric space associated to ${\displaystyle G}$ and ${\displaystyle T\subset G}$ is a maximal split torus then there exists a unique orbit of ${\displaystyle T}$ in ${\displaystyle X}$ which is a totally geodesic flat subspace in ${\displaystyle X}$. 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 ${\displaystyle X}$.[4]

If the Lie group ${\displaystyle G}$ is obtained as the real points of an algebraic group ${\displaystyle \mathbf {G} }$ over the rational field ${\displaystyle \mathbb {Q} }$ then the ${\displaystyle \mathbb {Q} }$-rank of ${\displaystyle \mathbf {G} }$ has also a geometric significance. To get to it one has to introduce an arithmetic group ${\displaystyle \Gamma }$ associated to ${\displaystyle \mathbf {G} }$, which roughly is the group of integer points of ${\displaystyle \mathbf {G} }$, and the quotient space ${\displaystyle M=\Gamma \backslash X}$, which is a Riemannian orbifold and hence a metric space. Then any asymptotic cone of ${\displaystyle M}$ is homeomorphic to a finite simplicial complex with top-dimensional simplices of dimension equal to the ${\displaystyle \mathbb {Q} }$-rank of ${\displaystyle \mathbf {G} }$. In particular, ${\displaystyle M}$ is compact if and only if ${\displaystyle \mathbf {G} }$ is anisotropic.[5]

Note that this allows to define the ${\displaystyle \mathbf {Q} }$-rank of any lattice in a semisimple Lie group, as the dimension of its asymptotic cone.

If ${\displaystyle \mathbf {G} }$ is a semisimple group over ${\displaystyle \mathbb {Q} _{p}}$ the maximal split tori in ${\displaystyle \mathbf {G} }$ correspond to the apartments of the Bruhat-Tits building ${\displaystyle X}$ associated to ${\displaystyle \mathbf {G} }$. In particular the dimension of ${\displaystyle X}$ is equal to the ${\displaystyle \mathbb {Q} _{p}}$-rank of ${\displaystyle \mathbf {G} }$.

Given a base scheme S, an algebraic torus over S is defined to be a group scheme 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 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.

One common example of an algebraic torus is to consider the affine cone ${\displaystyle {\text{Aff}}(X)\subset \mathbb {A} ^{n+1}}$ of a projective scheme ${\displaystyle X\subset \mathbb {P} ^{n}}$. Then, with the origin removed, the induced projection map

$${\displaystyle \pi$$ :({\text{Aff}}(X)-\{0\})\to X}

gives the structure of an algebraic torus over ${\displaystyle X}$.

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 ${\displaystyle H^{1}(S,GL_{n}(\mathbb {Z} ))}$, 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 ${\displaystyle H^{1}(G_{K},GL_{n}(\mathbb {Z} ))}$ 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 Ext¹ sheaves. These are naturally isomorphic to the flat cohomology groups ${\displaystyle H^{1}(S,\mathrm {Hom} _{\mathbb {Z} }(X^{\bullet }(T_{1}),X^{\bullet }(T_{2})))}$. Over a field, the extensions are parametrized by elements of the corresponding Galois cohomology group.