# 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) |

#### Comments

#### 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 |

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Rank of a Lie algebra.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Rank_of_a_Lie_algebra&oldid=48432