Maximal torus
A maximal torus of a linear algebraic group 
is an algebraic subgroup of 
which is an algebraic torus and which is not contained in any larger subgroup of that type. Now let 
be connected. The union of all maximal tori of 
coincides with the set of all semi-simple elements of 
(see Jordan decomposition) and their intersection coincides with the set of all semi-simple elements of the centre of 
. Every maximal torus is contained in some Borel subgroup of 
. The centralizer of a maximal torus is a Cartan subgroup of 
; it is always connected. Any two maximal tori of 
are conjugate in 
. If 
is defined over a field 
, then there is a maximal torus in 
also defined over 
; its centralizer is also defined over 
.
Let 
be a reductive group defined over a field 
. Consider the maximal subgroups among all algebraic subgroups of 
which are 
-split algebraic tori. The maximal 
-split tori thus obtained are conjugate over 
. The common dimension of these tori is called the 
-rank of 
and is denoted by 
. A maximal 
-split torus need not, in general, be a maximal torus, that is, 
is in general less than the rank of 
(which is equal to the dimension of a maximal torus in 
). If 
, then 
is called an anisotropic group over 
, and if 
coincides with the rank of 
, then 
is called a split group over 
. If 
is algebraically closed, then 
is always split over 
. In general, 
is split over the separable closure of 
.
Examples. Let 
be a field and let 
be an algebraic closure. The group 
of non-singular matrices of order 
with coefficients in 
(see Classical group; General linear group) is defined and split over the prime subfield of 
. The subgroup of all diagonal matrices is a maximal torus in 
.
Let the characteristic of 
be different from 2. Let 
be an 
-dimensional vector space over 
and 
a non-degenerate quadratic form on 
defined over 
(the latter means that in some basis 
of 
, the form 
is a polynomial in 
with coefficients in 
). Let 
be the group of all non-singular linear transformations of 
with determinant 1 and preserving 
. It is defined over 
. Let 
be the linear hull over 
of 
; it is a 
-form of 
. In 
there always exists a basis 
such that
 |
where 
if 
is even and 
if 
is odd. The subgroup of 
consisting of the elements whose matrix in this basis takes the form 
, where 
for 
and 
for 
, is a maximal torus in 
(thus the rank of 
is equal to the integer part of 
). In general, this basis does not belong to 
. However, there always is a basis 
in 
in which the quadratic form can be written as
 |
 |
where 
is a quadratic form which is anisotropic over 
(that is, the equation 
only has the zero solution in 
, see Witt decomposition). The subgroup of 
consisting of the elements whose matrix in the basis 
takes the form 
, where 
for 
, 
for 
and 
for 
, is a maximal 
-split torus in 
(so 
and 
is split if and only if 
is the integer part of 
).
Using maximal tori one associates to a reductive group 
a root system, which is a basic ingredient for the classification of reductive groups. Namely, let 
be the Lie algebra of 
and let 
be a fixed maximal torus in 
. The adjoint representation of 
in 
is rational and diagonalizable, so 
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 
, where 
is the group of rational characters of 
) turns out to be a (reduced) root system. The relative root system is defined in a similar way: If 
is defined over 
and 
is a maximal 
-split torus in 
, then the set of non-zero weights of the adjoint representation of 
in 
forms a root system (which need not be reduced) in some subspace of 
. See also Weyl group; Semi-simple group.
References
| [1] | A. Borel, "Linear algebraic groups" , Benjamin (1969) |
| [2] | A. Borel (ed.) G.D. Mostow (ed.) , Algebraic groups and discontinuous subgroups , Proc. Symp. Pure Math. , 9 , Amer. Math. Soc. (1966) |
Comments
For 
-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 
is a connected compact commutative Lie subgroup 
of 
not contained in any larger subgroup of the same type. As a Lie group 
is isomorphic to a direct product of copies of the multiplicative group of complex numbers of absolute value 1. Every maximal torus of 
is contained in a maximal compact subgroup of 
; any two maximal tori of 
(as any two maximal compact subgroups) are conjugate in 
. This, in a well-known sense, reduces the study of maximal tori to the case when 
is compact.
Now let 
be a compact group. The union of all maximal tori of 
is 
and their intersection is the centre of 
. The Lie algebra of a maximal torus 
is a maximal commutative subalgebra in the Lie algebra 
of 
, and each maximal commutative subalgebra in 
can be obtained in this way. The centralizer of a maximal torus 
in 
coincides with 
. The adjoint representation of 
in 
is diagonalizable and all non-zero weights of this representation form a root system in 
, where 
is the group of characters of 
. 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) |
| [2] | D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) |
| [3] | S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) |
Comments
References
| [a1] | N. Bourbaki, "Groupes et algèbres de Lie" , Eléments de mathématiques , Masson (1982) pp. Chapt. 9. Groupes de Lie réels compacts |
| [a2] | Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) |
Maximal torus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximal_torus&oldid=16122