Bivariant K-theory of generalized Weyl algebras

TL;DR

This paper computes the algebraic KK-theory classes of various noncommutative generalized Weyl algebras, including quantum Weyl algebras and quantum weighted projective lines, expanding understanding of their classification.

Contribution

It provides explicit KK-theory classifications for a broad class of noncommutative algebras, including quantum and primitive factor algebras, which was previously not fully understood.

Findings

Computed KK-theory classes for quantum Weyl algebra

Classified primitive factors of U(sl_2) in KK-theory

Analyzed quantum weighted projective lines in KK-theory

Abstract

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