Let . Assuming that is not a root of unity, we compute the Picard group of the algebra , describe its action on the space of isomorphism classes of rank 1 projective modules and classify the algebras Morita equivalent to . Our computations are based on a ‘quantum’ version of the Calogero–Moser correspondence relating projective -modules to irreducible representations of the double affine Hecke algebras at . We show that, under this correspondence, the action of on agrees with the action of on constructed by Cherednik [C1], [C2]. We compare our results with the smooth and analytic cases. In particular, when , we find that , where is the bounded derived category of coherent sheaves on the elliptic curve .
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. 335–356DOI 10.4171/JNCG/119