Rank of a Lie algebra

The minimal multiplicity of the eigen value $ \lambda = 0 $ for the linear operators $ \mathop{\rm ad} _ {L} x $, where $ x $ runs through the whole of the Lie algebra $ L $. It is assumed that $ L $ is a finite-dimensional algebra. An element $ x $ for which the multiplicity is minimal is called regular. The set of regular elements of a Lie algebra is open (in the Zariski topology). The rank of a Lie algebra is equal to the dimension of any Cartan subalgebra of it. The rank $ \mathop{\rm rk} L $ of a non-zero Lie algebra $ L $ satisfies the inequalities

$$ 1 \leq \mathop{\rm rk} L \leq \mathop{\rm dim} L, $$

and equality $ \mathop{\rm rk} L = \mathop{\rm dim} L $ holds if and only if $ L $ is nilpotent (cf. Lie algebra, nilpotent). For a semi-simple Lie algebra (cf. Lie algebra, semi-simple) over a field $ k $ the rank coincides with the transcendence degree over $ k $ of the subfield of the field of rational functions on $ L $ generated by all coefficients of the characteristic polynomials of the endomorphism $ \mathop{\rm ad} _ {L} x $.

If $ R $ is the radical in $ L $, then the rank of $ L / R $ is called the semi-simple rank of the algebra $ L $.

Examples. Let $ L $ be one of the following Lie algebras: 1) the algebra $ \mathfrak g \mathfrak l _ {n} $ of all square matrices of order $ n $ with elements from $ k $; 2) the algebra $ \mathfrak s \mathfrak l _ {n} $ of all matrices with zero trace; 3) the algebra of all upper-triangular matrices; 4) the algebra of all diagonal matrices; or 5) the algebra of all upper-triangular matrices with zeros on the principal diagonal. For these algebras the ranks are $ n $, $ n - 1 $, $ n $, $ n $, $ n ( n - 1 ) / 2 $, and the semi-simple ranks are $ n - 1 $, $ n - 1 $, $ 0 $, $ 0 $, $ 0 $.

References

[1]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[2]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[3]	C. Chevalley, "Théorie des groupes de Lie" , 3 , Hermann (1955)

References

[a1]	J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972)
[a2]	N. Bourbaki, "Eléments de mathématique. Groupes et algèbres de Lie" , Hermann (1975) pp. Chapt. 7