Symmetric quantum Weyl algebras

TL;DR

This paper investigates the symmetric powers of four key algebras, providing explicit formulas and combinatorial interpretations for their product normal coordinates, advancing understanding of their algebraic structures.

Contribution

It introduces explicit formulas and combinatorial interpretations for the normal coordinates of products in four important algebras, a novel contribution to algebraic combinatorics.

Findings

Explicit formulas for normal coordinates

Combinatorial interpretations provided

Enhanced understanding of algebraic structures

Abstract

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.