The Picard group of a noncommutative algebraic torus

  • Yuri Berest

    Cornell University, USA
  • Ajay C. Ramadoss

    ETH Zurich, Switzerland
  • Xiang Tang

    Washington University, St. Louis, USA


Let Aq:=Cx±1,y±1/(xyqyx)A_q := \mathbb{C}\langle x^{\pm 1}, y^{\pm 1}\rangle/(xy-qyx) . Assuming that qq is not a root of unity, we compute the Picard group Pic(Aq)\operatorname{Pic}(A_q) of the algebra AqA_q, describe its action on the space R(Aq)\mathcal{R}(A_q) of isomorphism classes of rank 1 projective modules and classify the algebras Morita equivalent to AqA_q . Our computations are based on a ‘quantum’ version of the Calogero–Moser correspondence relating projective AqA_q-modules to irreducible representations of the double affine Hecke algebras Ht,q1/2(Sn){\mathbb H}_{t, q^{-1/2}}(S_n) at t=1t = 1 . We show that, under this correspondence, the action of Pic(Aq)\operatorname{Pic}(A_q) on R(Aq)\mathcal{R}(A_q) agrees with the action of SL2(Z)\operatorname{SL}_2(\mathbb{Z}) on Ht,q1/2(Sn){\mathbb H}_{t, q^{-1/2}}(S_n) constructed by Cherednik [C1], [C2]. We compare our results with the smooth and analytic cases. In particular, when q1|q| \ne 1 , we find that Pic(Aq)Auteq(Db(X))/Z\operatorname{Pic}(A_q) \cong \operatorname{Auteq} (\mathscr{D}^{\mathrm{b}}(X))/{\mathbb{Z}} , where Db(X)\mathscr{D}^{\mathrm{b}}(X) is the bounded derived category of coherent sheaves on the elliptic curve X=C/ZX = \mathbb{C}^*/ {\mathbb{Z}} .

Cite this article

Yuri Berest, Ajay C. Ramadoss, Xiang Tang, The Picard group of a noncommutative algebraic torus. J. Noncommut. Geom. 7 (2013), no. 2, pp. 335–356

DOI 10.4171/JNCG/119