# Maximal torus

A maximal torus of a linear algebraic group $G$ is an algebraic subgroup of $G$ which is an algebraic torus and which is not contained in any larger subgroup of that type. Now let $G$ be connected. The union of all maximal tori of $G$ coincides with the set of all semi-simple elements of $G$ ( see Jordan decomposition) and their intersection coincides with the set of all semi-simple elements of the centre of $G$ . Every maximal torus is contained in some Borel subgroup of $G$ . The centralizer of a maximal torus is a Cartan subgroup of $G$ ; it is always connected. Any two maximal tori of $G$ are conjugate in $G$ . If $G$ is defined over a field $k$ , then there is a maximal torus in $G$ also defined over $k$ ; its centralizer is also defined over $k$ .

Let $G$ be a reductive group defined over a field $k$ . Consider the maximal subgroups among all algebraic subgroups of $G$ which are $k$ - split algebraic tori. The maximal $k$ - split tori thus obtained are conjugate over $k$ . The common dimension of these tori is called the $k$ - rank of $G$ and is denoted by $\mathop{\rm rk}\nolimits _{k} \ G$ . A maximal $k$ - split torus need not, in general, be a maximal torus, that is, $\mathop{\rm rk}\nolimits _{k} \ G$ is in general less than the rank of $G$ ( which is equal to the dimension of a maximal torus in $G$ ). If $\mathop{\rm rk}\nolimits _{k} \ G = 0$ , then $G$ is called an anisotropic group over $k$ , and if $\mathop{\rm rk}\nolimits _{k} \ G$ coincides with the rank of $G$ , then $G$ is called a split group over $k$ . If $k$ is algebraically closed, then $G$ is always split over $k$ . In general, $G$ is split over the separable closure of $k$ .

Examples. Let $k$ be a field and let $\overline{k}$ be an algebraic closure. The group $G = \mathop{\rm GL}\nolimits _{n} ( \overline{k} )$ of non-singular matrices of order $n$ with coefficients in $\overline{k}$ ( see Classical group; General linear group) is defined and split over the prime subfield of $k$ . The subgroup of all diagonal matrices is a maximal torus in $G$ .

Let the characteristic of $k$ be different from 2. Let $V$ be an $n$ - dimensional vector space over $\overline{k}$ and $F$ a non-degenerate quadratic form on $V$ defined over $k$ ( the latter means that in some basis $e _{1} \dots e _{n}$ of $V$ , the form $F ( x _{1} e _{1} + {} \dots + x _{n} e _{n} )$ is a polynomial in $x _{1} \dots x _{n}$ with coefficients in $k$ ). Let $G$ be the group of all non-singular linear transformations of $V$ with determinant 1 and preserving $F$ . It is defined over $k$ . Let $V _{k}$ be the linear hull over $k$ of $e _{1} \dots e _{n}$ ; it is a $k$ - form of $V$ . In $V$ there always exists a basis $f _{1} \dots f _{n}$ such that $$F ( x _{1} f _{1} + \dots + x _{n} f _{n} ) = x _{1} x _{n} + x _{2} x _{n-1} + \dots + x _{p} x _{n-p+1} ,$$ where $p = n / 2$ if $n$ is even and $p = ( n + 1 ) / 2$ if $n$ is odd. The subgroup of $G$ consisting of the elements whose matrix in this basis takes the form $\| a _{ij} \|$ , where $a _{ij} = 0$ for $i \neq j$ and $a _{ii} a _{n-i+1},n-i+1 = 1$ for $i = 1 \dots p$ , is a maximal torus in $G$ ( thus the rank of $G$ is equal to the integer part of $n / 2$ ). In general, this basis does not belong to $V _{k}$ . However, there always is a basis $h _{1} \dots h _{n}$ in $V _{k}$ in which the quadratic form can be written as $$F ( x _{1} h _{1} + \dots + x _{n} h _{n} ) =$$ $$= x _{1} x _{n} + \dots + x _{q} x _{n-q+1} + F _{0} ( x _{q+1} \dots x _{n-q} ) , q > p ,$$ where $F _{0}$ is a quadratic form which is anisotropic over $k$ ( that is, the equation $F _{0} = 0$ only has the zero solution in $k$ , see Witt decomposition). The subgroup of $G$ consisting of the elements whose matrix in the basis $h _{1} \dots h _{n}$ takes the form $\| a _{ij} \|$ , where $a _{ij} = 0$ for $i \neq j$ , $a _{ii} a _{n-i+1},n-i+1 = 1$ for $i = 1 \dots q$ and $a _{ii} = 1$ for $i = q + 1 \dots n - q$ , is a maximal $k$ - split torus in $G$ ( so $\mathop{\rm rk}\nolimits _{k} \ G = q$ and $G$ is split if and only if $q$ is the integer part of $n / 2$ ).

Using maximal tori one associates to a reductive group $G$ a root system, which is a basic ingredient for the classification of reductive groups. Namely, let $\mathfrak g$ be the Lie algebra of $G$ and let $T$ be a fixed maximal torus in $G$ . The adjoint representation of $T$ in $\mathfrak g$ is rational and diagonalizable, so $\mathfrak g$ decomposes into a direct sum of weight spaces for this representation. The set of non-zero weights of this representation (considered as a subset of its linear hull in the vector space $X (T) \otimes _{\mathbf Z} \mathbf R$ , where $X (T)$ is the group of rational characters of $T$ ) turns out to be a (reduced) root system. The relative root system is defined in a similar way: If $G$ is defined over $k$ and $S$ is a maximal $k$ - split torus in $G$ , then the set of non-zero weights of the adjoint representation of $S$ in $\mathfrak g$ forms a root system (which need not be reduced) in some subspace of $X (S) \otimes _{\mathbf Z} \mathbf R$ . See also Weyl group; Semi-simple group.

## Contents

#### References

 [1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201 [2] A. Borel (ed.) G.D. Mostow (ed.) , Algebraic groups and discontinuous subgroups , Proc. Symp. Pure Math. , 9 , Amer. Math. Soc. (1966) MR0202512 Zbl 0171.24105

For $k$ - forms see Form of an (algebraic) structure.

See especially the article by A. Borel in [2].

A maximal torus of a connected real Lie group $G$ is a connected compact commutative Lie subgroup $T$ of $G$ not contained in any larger subgroup of the same type. As a Lie group $T$ is isomorphic to a direct product of copies of the multiplicative group of complex numbers of absolute value 1. Every maximal torus of $G$ is contained in a maximal compact subgroup of $G$ ; any two maximal tori of $G$ ( as any two maximal compact subgroups) are conjugate in $G$ . This, in a well-known sense, reduces the study of maximal tori to the case when $G$ is compact.

Now let $G$ be a compact group. The union of all maximal tori of $G$ is $G$ and their intersection is the centre of $G$ . The Lie algebra of a maximal torus $T$ is a maximal commutative subalgebra in the Lie algebra $\mathfrak g$ of $G$ , and each maximal commutative subalgebra in $\mathfrak g$ can be obtained in this way. The centralizer of a maximal torus $T$ in $G$ coincides with $T$ . The adjoint representation of $T$ in $\mathfrak g$ is diagonalizable and all non-zero weights of this representation form a root system in $X (T) \otimes _{\mathbf Z} \mathbf R$ , where $X (T)$ is the group of characters of $T$ . This is a basic ingredient for the classification of compact Lie groups.

#### References

 [1] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104 [2] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013 [3] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) MR0514561 Zbl 0451.53038