We show that free objects on sets do not exist in the category \({\varvec{ba}}\varvec{\ell }\) of bounded archimedean \(\ell \)-algebras. On other hand, we introduce weighted and prove }\). conclude by discussing several consequences this result.

Let K be a field of characteristic zero. For integers n,D≥1, let Jn(D) the degree n component free Jordan algebra J(D) over D generators. A conjecture for character (in particular dimension) GL(D)-module is proposed. sl2J(D) Tits-Allison-Gao construction J(D). Two natural conjectures homology Lie are stated, and each them implies previous conjecture. The cyclicity structures, namely that symmet...