Rank of a Lie algebra

The minimal multiplicity of the eigen value for the linear operators , where runs through the whole of the Lie algebra . It is assumed that is a finite-dimensional algebra. An element 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 of a non-zero Lie algebra satisfies the inequalities

and equality holds if and only if is nilpotent (cf. Lie algebra, nilpotent). For a semi-simple Lie algebra (cf. Lie algebra, semi-simple) over a field the rank coincides with the transcendence degree over of the subfield of the field of rational functions on generated by all coefficients of the characteristic polynomials of the endomorphism .

If is the radical in , then the rank of is called the semi-simple rank of the algebra .

Examples. Let be one of the following Lie algebras: 1) the algebra of all square matrices of order with elements from ; 2) the algebra 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 , , , , , and the semi-simple ranks are , , , , .

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