# Rank of an algebraic group

Jump to: navigation, search

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 $G$ .

How to Cite This Entry:
Rank of an algebraic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rank_of_an_algebraic_group&oldid=44282
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article