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

中文翻译：

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广义Weyl代数的双变量K理论

我们在$ \ mathfrak {KK} ^ {\ mathrm {alg}} $中计算所有非交换广义Weyl代数$ A = \ mathbb C [h]（\ sigma，P）$的同构类，其中$ \ sigma（h ）= qh + h_0 $是$ \ mathcal C [h] $的自同构，除非$ q \ neq 1 $是单位根。具体来说，我们计算量子Weyl代数的$ \ mathfrak {KK} ^ {\ mathrm {alg}} $中的同构类，即$ U（\ mathfrak {sl} _2 ）$和量子加权投影线$ \ mathcal {O}（\ mathbb {WP} _q（k，l））$。