The Picard Group of a Noncommutative Algebraic Torus

TL;DR

This paper computes the Picard group of a noncommutative algebraic 2-torus, relates it to representations of double affine Hecke algebras, and classifies Morita equivalences, revealing deep connections with elliptic curves and quantum symmetries.

Contribution

It provides the first detailed computation of the Picard group for noncommutative algebraic tori and links it to DAHA representations and derived category auto-equivalences.

Findings

Picard group is isomorphic to auto-equivalences of derived category modulo translations.

Action of Picard group matches SL_2(Z) action on DAHA.

Results extend to smooth and analytic cases, especially when |q| ≠ 1.

Abstract

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. In particular, when $ |q| \not= 1 $, we find that $ Pic(A_q) $ is isomorphic to the group of auto-equivalences $ Auteq(D^b(X))/Z $ of the bounded derived category of coherent sheaves on the elliptic curve $ X = C*/Z $ modulo translations.