Bivariant K-theory of generalized Weyl algebras

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