A note on centralizers in \(q\)-deformed Heisenberg algebras (Q2723255)

Let \(A\) be a generalized Weyl algebra over a field \(k\) generated by elements \(X,Y,H\) subject to the defining relations NEWLINE\[NEWLINEYX=t(H)\in k[H],\quad XY=t(qH+1),\quad Xa(H)=a(qH+1)X,\quad a(H)Y=Ya(qH+1),NEWLINE\]NEWLINE for all \(a(H)\in k[H]\), where \(q\in k^*\). It is shown that if \(q\in k^*\) is not a root of 1 then the centralizer \(C(f)\) of any non-scalar element \(f\in A\) is a commutative algebra and a free \(k[f]\)-module of finite rank. In particular, any two commuting elements of the \(q\)-deformed Heisenberg algebra \({\mathcal H}(q)\) are algebraically dependent. It is observed that if \(q\) is a root of 1 then \(A\) is a finitely generated module over its center.