The Picard group of a noncommutative algebraic torus

Abstract

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 $\operatorname{Pic}(A_q)$ 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 $\operatorname{Pic}(A_q)$ on $\mathcal{R}(A_q)$ agrees with the action of $\operatorname{SL}_2(\mathbb{Z})$ on ${\mathbb H}_{t, q^{-1/2}}(S_n)$ constructed by Cherednik [C1], [C2]. We compare our results with the smooth and analytic cases. In particular, when $|q| \ne 1$ , we find that $\operatorname{Pic}(A_q) \cong \operatorname{Auteq} (\mathscr{D}^{\mathrm{b}}(X))/{\mathbb{Z}}$ , where $\mathscr{D}^{\mathrm{b}}(X)$ is the bounded derived category of coherent sheaves on the elliptic curve $X = \mathbb{C}^*/ {\mathbb{Z}}$ .