Invariant Indentities in the Heisenberg Algebra

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 $n$ \[ (aba)^n = a^n b^n a^n \]