The Picard group of a noncommutative algebraic torus. (Q355356)

Summary: Let \(A_q:=\mathbb C\langle x^{\pm 1},y^{\pm 1}\rangle/(xy-qyx)\). Assuming that \(q\) is not a root of unity, we compute the Picard group \(\mathrm{Pic}(A_q)\) of the algebra \(A_q\), describe its action on the space \(\mathcal R(A_q)\) of isomorphism classes of rank 1 projective modules and classify the algebras Morita equivalent to \(A_q\). Our computations are based on a `quantum' version of the Calogero-Moser correspondence relating projective \(A_q\)-modules to irreducible representations of the double affine Hecke algebras \(\mathbb H_{t,q^{-1/2}}(S_n)\) at \(t=1\). We show that, under this correspondence, the action of \(\mathrm{Pic}(A_q)\) on \(\mathcal R(A_q)\) agrees with the action of \(\mathrm{SL}_2(\mathbb Z)\) on \(\mathbb H_{t,q^{-1/2}}(S_n)\) constructed by \textit{I. Cherednik} [Int. Math. Res. Not. 1992, No. 9, 171-180 (1992; Zbl 0770.17004); Invent. Math. 122, No. 1, 119-145 (1995; Zbl 0854.22021)]. We compare our results with the smooth and analytic cases. In particular, when \(|q|\neq 1\), we find that \(\mathrm{Pic}(A_q)\cong\mathrm{Auteq}(\mathcal D^{\mathrm b}(X))/\mathbb Z\), where \(\mathcal D^{\mathrm b}(X)\) is the bounded derived category of coherent sheaves on the elliptic curve \(X=\mathbb C^*/\mathbb Z\).