Bivariant K-theory of generalized Weyl algebras

Abstract

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 $q\ne 1$ 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 $\mathcal{O}({\mathbb{WP}}_q(k,l))$.