The Picard Group of a Noncommutative Algebraic Torus

2010/10/19 by Yuri Berest, Berest, Yuri, Ajay C. Ramadoss +4
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #K-Theory and Homology (math.KT) #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math-ph #math.KT #math.MP #math.QA #math.RT

paper · pdf · doi:10.48550/arxiv.1010.3779

15 pages

arxiv created 2010/10/19 · openalex publication_date 2010/10/19 · arxiv updated 2010/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 SL₂(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.

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