# The Picard group of a noncommutative algebraic torus

### Yuri Berest

Cornell University, USA### Ajay C. Ramadoss

ETH Zurich, Switzerland### Xiang Tang

Washington University, St. Louis, USA

## Abstract

Let $A_{q}:=C⟨x_{±1},y_{±1}⟩/(xy−qyx)$ . Assuming that $q$ is not a root of unity, we compute the Picard group $Pic(A_{q})$ of the algebra $A_{q}$, describe its action on the space $R(A_{q})$ 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 $H_{t,q_{−1/2}}(S_{n})$ at $t=1$ . We show that, under this correspondence, the action of $Pic(A_{q})$ on $R(A_{q})$ agrees with the action of $SL_{2}(Z)$ on $H_{t,q_{−1/2}}(S_{n})$ constructed by Cherednik [C1], [C2]. We compare our results with the smooth and analytic cases. In particular, when $∣q∣=1$ , we find that $Pic(A_{q})≅Auteq(D_{b}(X))/Z$ , where $D_{b}(X)$ is the bounded derived category of coherent sheaves on the elliptic curve $X=C_{∗}/Z$ .

## Cite this article

Yuri Berest, Ajay C. Ramadoss, Xiang Tang, The Picard group of a noncommutative algebraic torus. J. Noncommut. Geom. 7 (2013), no. 2, pp. 335–356

DOI 10.4171/JNCG/119