The Picard group of a noncommutative algebraic torus

Abstract

Let $A_q:=\mathbb{C}⟨x^{\pm 1},y^{\pm 1}⟩/(xy-qyx)$ . 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 Cherednik [C1], [C2]. We compare our results with the smooth and analytic cases. In particular, when $|q|\ne 1$ , 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 $X={\mathbb{C}}^{\ast }/\mathbb{Z}$ .