The Picard Group of a Noncommutative Algebraic Torus

Understand

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 $ SL_2(Z) $ on $ H_{t, q^{-1/2}}(S_n) $ constructed by I.Cherednik.
• We compare our results with smooth and analytic cases.