The Picard group of a noncommutative algebraic torus

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,H-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 SL2 (Z) on H-t,H-q-1/2 (S-n) constructed by Cherednik [C1], [C2]. We compare our results with the smooth and analytic cases. In particular, when vertical bar q vertical bar not equal 1, we find that Pic(A(q)) congruent to Auteq(D-b(X))/Z, where D-b(X) is the bounded derived category of coherent sheaves on the elliptic curve X = C*/Z.