# 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.

#### 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 |

#### Comments

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 |

#### Comments

#### References

[a1] | N. Bourbaki, "Groupes et algèbres de Lie" , Eléments de mathématiques , Masson (1982) pp. Chapt. 9. Groupes de Lie réels compacts MR0682756 Zbl 0505.22006 |

[a2] | Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) MR0781344 Zbl 0581.22009 |

**How to Cite This Entry:**

Maximal torus.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Maximal_torus&oldid=44278