The Picard Group of a Noncommutative Algebraic Torus

We compute the Picard group $Pic(A_q)$ of the noncommutative algebraic 2-torus $A_q$, describe its action on the space $R(A_q)$ of isomorphism classes of rk 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 (DAHA) $ 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 $S{L}_2(Z)$ on $H_{t,q^{-1\mathrm{/}2}}(S_n)$ constructed by I.Cherednik. We compare our results with smooth and analytic cases. In particular, when $\mathrm{\mid }q\mathrm{\mid }≠1$, we find that $Pic(A_q)$ is isomorphic to the group of auto-equivalences $Auteq(D^b(X))\mathrm{/}Z$ of the bounded derived category of coherent sheaves on the elliptic curve $ X = C*/Z $ modulo translations.