 
Summary: MATH. SCAND. 71 (1991),15..20
ROOT MULTIPLICITIES AND IDEALS IN
QUASISIMPLE LIE ALGEBRAS
HELMER ASLAKSEN, TERJE WAHL and TORE WENTZELLARSEN
1. Introduction.
The purpose of this note is to elaborate on some of the results of a paper by
H0eghKrohn and Torresani [1]. They determined the possible root systems for
quasisimple Lie algebras, but they did not discuss the multiplicities of the roots.
The nonisotropic roots can only occur with multiplicity one, and the multiplic
ities of isotropic roots in the affine Lie algebras are known. But for general
quasisimple Lie algebras, the multiplicities of the isotropic roots are more
complicated. We call the dimension ofthe span of the isotropic roots the type and
denote it by v. We will give examples ofquasisimple Lie algebras oftype two with
the same sets of roots but with different multiplicities. This shows that although
H0eghKrohn and Torresani have determined the possible root systems (not
counting multiplicities) and given explicit realizations for the root systems oftype
v ~ 2 it does not follow that we know all quasisimple Lie algebras with v ~ 2.
There is also the question of whether the root system determines the Lie
