1. CJM Online first
||Order $3$ elements in $G_2$ and idempotents in symmetric composition algebras|
Order three elements in the exceptional groups of type $G_2$
are classified up to conjugation over arbitrary fields. Their
centralizers are computed, and the associated classification
of idempotents in symmetric composition algebras is obtained.
Idempotents have played a key role in the study and classification
of these algebras.
Over an algebraically closed field, there are two conjugacy classes
of order three elements in $G_2$ in characteristic not $3$ and
four of them in characteristic $3$. The centralizers in characteristic
$3$ fail to be smooth for one of these classes.
Keywords:symmetric composition algebra, Okubo algebra, automorphism group, centralizer, idempotent
Categories:17A75, 14L15, 17B25, 20G15
2. CJM 1998 (vol 50 pp. 225)
||Derivations and invariant forms of Lie algebras graded by finite root systems |
Lie algebras graded by finite reduced root systems have been
classified up to isomorphism. In this paper we describe the derivation
algebras of these Lie algebras and determine when they possess invariant
bilinear forms. The results which we develop to do this are much more
general and apply to Lie algebras that are completely reducible with
respect to the adjoint action of a finite-dimensional subalgebra.
Categories:17B20, 17B70, 17B25