The Picard group of a noncommutative algebraic torus
Y Berest, AC Ramadoss, X Tang - Journal of Noncommutative Geometry, 2013 - ems.press
Let Aq Chx 1; y 1i=. xy qyx/. Assuming that q is not a root of unity, we compute the Picard
group Pic. Aq/of the algebra Aq, describe its action on the space RA q/of isomorphism
classes of rank 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 Ht; q 1= 2. Sn/at t D 1. We show that, under this correspondence, the action of Pic …
group Pic. Aq/of the algebra Aq, describe its action on the space RA q/of isomorphism
classes of rank 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 Ht; q 1= 2. Sn/at t D 1. We show that, under this correspondence, the action of Pic …
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