2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/98564We study the symmetric powers of four algebras: $q$-oscillator algebra, $q$-Weyl algebra, $h$-Weyl algebra and $U({\mathfrak {sl}}_2)$. We provide explicit formulae as well as combinatorial interpretation for the normal coordinates of products of arbitrary elements in the above algebras.To appear in the Annales Mathematiques Blaise PascalQuantum AlgebraHigh Energy Physics - TheoryQuantum Physicstext