The Picard Group of a Noncommutative Algebraic Torus

Yuri Berest; Ajay Ramadoss; Xiang Tang

paper

The Picard Group of a Noncommutative Algebraic Torus

arXiv:1010.3779

Abstract

We compute the Picard group $P i c (A_{q})$ 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 $P i c (A_{q})$ on $R (A_{q})$ agrees with the action of $S L_{2} (Z)$ on $H_{t, q^{- 1/2}} (S_{n})$ constructed by I.Cherednik. We compare our results with smooth and analytic cases. In particular, when $∣ q ∣ \neq = 1$ , we find that $P i c (A_{q})$ is isomorphic to the group of auto-equivalences $A u t e q (D^{b} (X)) / Z$ of the bounded derived category of coherent sheaves on the elliptic curve $X = C * / Z$ modulo translations.

15 pages