# 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)