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) 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 -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) 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

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