The Picard Group of a Noncommutative Algebraic Torus

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We compute the Picard group Pic(Aq) of the noncommutative algebraic 2-torus Aq, describe its action on the space R(Aq) of isomorphism classes of rk 1 projective modules and classify the algebras Morita equivalent to Aq. Our computations are based on a quantum version of the Calogero-Moser correspondence relating projective Aq-modules to irreducible representations of the double affine Hecke algebras (DAHA) Hₜ, q⁻¹/²(Sₙ) at t = 1. We show that, under this correspondence, the action of Pic(Aq) on R(Aq) agrees with the action of SL₂(Z) on Hₜ, q⁻¹/²(Sₙ) constructed by I.Cherednik. We compare our results with smooth and analytic cases. In particular, when |q| = 1, we find that Pic(Aq) is isomorphic to the group of auto-equivalences Auteq(Dᵇ(X))/Z of the bounded derived category of coherent sheaves on the elliptic curve X = C*/Z modulo translations.