Rank of an algebraic group

The dimension of a Cartan subgroup of it (this dimension does not depend on the choice of the Cartan subgroup). Along with the rank of an algebraic group $ G $ one considers its semi-simple rank and reductive rank, which, by definition, are equal to the rank of the algebraic group $ G / R $ and the rank of the algebraic group $ G / R _{u} $ respectively, where $ R $ is the radical of the algebraic group $ G $ and $ R _{u} $ is its unipotent radical (cf. Radical of a group; Unipotent element). The reductive rank of an algebraic group $ G $ is equal to the dimension of any of its maximal tori (cf. Maximal torus). The reductive $ k $ - rank of a linear algebraic group $ G $ defined over a field $ k $ ( and in the case when the group $ G $ is reductive (cf. Reductive group) — simply its $ k $ - rank) is the dimension of a maximal $ k $ - split torus of it (this dimension does not depend on the choice of a torus; see Split group). If the $ k $ - rank of a reductive linear algebraic group $ G $ over $ k $ is zero (is equal to the rank of $ G $ ), then the group $ G $ is said to be anisotropic (or split, respectively) over $ k $ ( see also Anisotropic group).

Examples.

1) The rank of the algebraic group $ T _{n} $ of all non-singular upper-triangular square matrices of order $ n $ is equal to its reductive rank and equal to $ n $ ; the semi-simple rank of $ T _{n} $ is zero.

2) The rank of the algebraic group $ U _{n} $ of all upper-triangular square matrices of order $ n $ with 1 on the principal diagonal is equal to its dimension $ n ( n - 1 ) / 2 $ , and the reductive and semi-simple ranks of $ U _{n} $ are zero.

3) The rank of the algebraic group $ O _{n} ( k ,\ f \ ) $ of all $ k $ - automorphisms of a definite quadratic form $ f $ on an $ n $ - dimensional vector space over a field $ k $ is equal to $ [ n / 2 ] $ , and the $ k $ - rank of the group $ O _{n} ( k ,\ f \ ) $ is equal to the Witt index of the form $ f $ .

If the characteristic of the ground field is 0, then the rank of the algebraic group $ G $ coincides with the rank of its Lie algebra $ L $ ( see Rank of a Lie algebra) and is equal to the minimum multiplicity of the eigen value $ \lambda = 1 $ of all possible adjoint operators $ \mathop{\rm Ad}\nolimits _{L} \ g $ ( the minimum is taken over all $ g \in G $ ). An element $ g \in G $ for which this multiplicity is equal to the rank of the algebraic group $ G $ is called regular. The set of regular elements of $ G $ is open in the Zariski topology on $ G $ .

References

[1]	C. Chevalley, "Théorie des groupes de Lie" , 2–3 , Hermann (1952–1955) MR0068552 MR0051242 MR0019623 Zbl 0186.33104 Zbl 0054.01303 Zbl 0063.00843
[2]	A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–250 MR0207712 Zbl 0145.17402
[3]	A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[4]	J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039