Le support de l'algèbre de Lie libre. (Support of the free Lie algebra) (Q912194)

Let \({\mathbb{Z}}<X>\) be a free associative ring with a base X. The ring \({\mathbb{Z}}<X>\) is a Lie ring relative to the multiplication \([a,b]=ab-ba\). Denote by L(X) a Lie ring generated by X in \({\mathbb{Z}}<X>\). Then L(X) is a free Lie ring with base X. There exists a unique involution * in the ring \({\mathbb{Z}}<X>\) such that \(x*=x\) for all \(x\in X.\) Theorem. Let f be a monomial in \({\mathbb{Z}}<X>\). Then the following are equivalent: (i) f does not occur in any element from L(X); (ii) either \(f=x^ n\), \(n\geq 2\), \(x\in X\), or \(f=f*\) and f has even length.