User:Maximilian Janisch/latexlist/Algebraic Groups/Rank of an algebraic group

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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 $k$ 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 $G / R$ respectively, where $R$ is the radical of the algebraic group $k$ and $R$ is its unipotent radical (cf. Radical of a group; Unipotent element). The reductive rank of an algebraic group $k$ is equal to the dimension of any of its maximal tori (cf. Maximal torus). The reductive $k$-rank of a linear algebraic group $k$ defined over a field $k$ (and in the case when the group $k$ 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 $k$ over $k$ is zero (is equal to the rank of $k$), then the group $k$ 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 $12$ is equal to its reductive rank and equal to $12$; 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 $12$ 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 $O _ { x } ( k , f )$ of all $k$-automorphisms of a definite quadratic form $f$ on an $12$-dimensional vector space over a field $k$ is equal to $[ n / 2 ]$, and the $k$-rank of the group $O _ { x } ( k , f )$ 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 $k$ 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 $_ { L } g$ (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 $k$ is called regular. The set of regular elements of $k$ is open in the Zariski topology on $k$.

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