- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:13
Journal of Noncommutative Geometry
J. Noncommut. Geom.
JNCG
1661-6952
1661-6960
Global analysis, analysis on manifolds
General
10.4171/JNCG
http://www.ems-ph.org/doi/10.4171/JNCG
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
7
2013
2
The Picard group of a noncommutative algebraic torus
Yuri
Berest
Cornell University, ITHACA, UNITED STATES
Ajay
Ramadoss
ETH Zürich, ZÜRICH, SWITZERLAND
Xiang
Tang
Washington University, ST. LOUIS, UNITED STATES
Noncommutative algebraic torus, quantum Weyl algebra, projective module, Morita equivalence, Picard group, double affine Hecke algebra
Let $ A_q := \mathbb{C}\langle x^{\pm 1}, y^{\pm 1}\rangle/(xy-qyx) $. Assuming that $q$ is not a root of unity, we compute the Picard group $ \operatorname{Pic}(A_q) $ of the algebra $A_q$, describe its action on the space $ \mathcal{R}(A_q) $ of isomorphism classes of rank 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 $ {\mathbb H}_{t, q^{-1/2}}(S_n) $ at $ t = 1 $. We show that, under this correspondence, the action of $ \operatorname{Pic}(A_q) $ on $\mathcal{R}(A_q) $ agrees with the action of $\operatorname{SL}_2(\mathbb{Z}) $ on $ {\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 $ |q| \ne 1 $, we find that $ \operatorname{Pic}(A_q) \cong \operatorname{Auteq} (\mathscr{D}^{\mathrm{b}}(X))/{\mathbb{Z}} $, where $ \mathscr{D}^{\mathrm{b}}(X) $ is the bounded derived category of coherent sheaves on the elliptic curve $ X = \mathbb{C}^*/ {\mathbb{Z}} $.
Associative rings and algebras
General
335
356
10.4171/JNCG/119
http://www.ems-ph.org/doi/10.4171/JNCG/119