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

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