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 
one considers its semi-simple rank and reductive rank, which, by definition, are equal to the rank of the algebraic group 
and the rank of the algebraic group 
respectively, where 
is the radical of the algebraic group 
and 
is its unipotent radical (cf. Radical of a group; Unipotent element). The reductive rank of an algebraic group 
is equal to the dimension of any of its maximal tori (cf. Maximal torus). The reductive 
-rank of a linear algebraic group 
defined over a field 
(and in the case when the group 
is reductive (cf. Reductive group) — simply its 
-rank) is the dimension of a maximal 
-split torus of it (this dimension does not depend on the choice of a torus; see Split group). If the 
-rank of a reductive linear algebraic group 
over 
is zero (is equal to the rank of 
), then the group 
is said to be anisotropic (or split, respectively) over 
(see also Anisotropic group).
Examples.
1) The rank of the algebraic group 
of all non-singular upper-triangular square matrices of order 
is equal to its reductive rank and equal to 
; the semi-simple rank of 
is zero.
2) The rank of the algebraic group 
of all upper-triangular square matrices of order 
with 1 on the principal diagonal is equal to its dimension 
, and the reductive and semi-simple ranks of 
are zero.
3) The rank of the algebraic group 
of all 
-automorphisms of a definite quadratic form 
on an 
-dimensional vector space over a field 
is equal to 
, and the 
-rank of the group 
is equal to the Witt index of the form 
.
If the characteristic of the ground field is 0, then the rank of the algebraic group 
coincides with the rank of its Lie algebra 
(see Rank of a Lie algebra) and is equal to the minimum multiplicity of the eigen value 
of all possible adjoint operators 
(the minimum is taken over all 
). An element 
for which this multiplicity is equal to the rank of the algebraic group 
is called regular. The set of regular elements of 
is open in the Zariski topology on 
.
References
| [1] | C. Chevalley, "Théorie des groupes de Lie" , 2–3 , Hermann (1952–1955) |
| [2] | A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–250 |
| [3] | A. Borel, "Linear algebraic groups" , Benjamin (1969) |
| [4] | J.E. Humphreys, "Linear algebraic groups" , Springer (1975) |
Rank of an algebraic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rank_of_an_algebraic_group&oldid=21916