Bivariant K-theory of generalized Weyl algebras

We compute the isomorphism class in ${\mathfrak{K}\mathfrak{K}}^{alg}$ of all noncommutative generalized Weyl algebras $A=\text{CC}[h](\sigma ,P)$, where $\sigma (h)=qh+h_0$ is an automorphism of $\text{CC}[h]$, except when $q\mathrm{\ne }1$ is a root of unity. In particular, we compute the isomorphism class in ${\mathfrak{K}\mathfrak{K}}^{alg}$ of the quantum Weyl algebra, the primitive factors $B_{\lambda }$ of $U({\mathfrak{s}\mathfrak{l}}_2)$ and the quantum weighted projective lines $\mathcal{O}({\mathbb{W}\mathbb{P}}_q(k,l))$.