Invariant Indentities in the Heisenberg Algebra

TL;DR

This paper discovers polynomial relations invariant under quantization and q-deformation in the Heisenberg algebra, revealing new algebraic identities involving generators obeying specific q-commutation relations.

Contribution

It introduces new invariant polynomial relations in the q-deformed Heisenberg algebra, expanding understanding of algebraic structures under deformation.

Findings

Polynomial relations invariant under q-deformation are identified.

A specific identity for elements satisfying a q-commutation relation is proven.

The identities hold for all complex parameters p, q and natural n.

Abstract

Polynomial relations between the generators of $q$--deformed Heisenberg algebra invariant under the quantization and $q$-deformation are discovered. One of the examples of such relations is the following: if two elements $a$ and $b$, obeying the relation \[ ab - q ba = p, \] where $p, q$ are any complex numbers, then for any $p,q$ and natural $n$ \[ (aba)^n = a^n b^n a^n \]