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