Symmetric quantum Weyl algebras

We 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.Comment: To appear in the Annales Mathematiques Blaise Pasca