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Representations of quantum tori and \(G\)-bundles of elliptic curves. (English) Zbl 1161.17310

Duval, Christian (ed.) et al., The orbit method in geometry and physics. In honor of A. A. Kirillov. Papers from the international conference, Marseille, France, December 4–8, 2000. Boston, MA: Birkhäuser (ISBN 0-8176-4232-3/hbk). Prog. Math. 213, 29-48 (2003).
From the text: We study a BGG-type category of infinite-dimensional representations of \(\mathcal H[W]\), a semidirect product of the quantum torus with parameter \(q\), built on the root lattice of a semisimple group \(G\), and the Weyl group of \(G\). Irreducible objects of our category turn out to be parametrized by semistable \(G\)-bundles on the elliptic curve \(\mathbb C^*/q^\mathbb Z\).
We introduce a noncommutative deformation of the algebra of regular functions on a torus. This deformation \(\mathcal H\), called quantum torus algebra, depends on a complex parameter \(q\in\mathbb C^*\). We further introduce a certain category \(\mathcal M(\mathcal H,\mathcal A)\) of representations of \(\mathcal H\) which are locally-finite with respect to a commutative subalgebra \(\mathcal A\subset\mathcal H\) whose “size” is one-half of that of \(\mathcal H\) (our definition is modeled on the definition of the category \(\mathcal O\) of Bernstein-Gelfand-Gelfand). We classify all simple objects of \(\mathcal M(\mathcal H,\mathcal A)\) and show that any object of \(\mathcal M(\mathcal H,\mathcal A)\) has finite length.
In §3 we consider quantum tori arising from a pair of lattices coming from a finite reduced root system. Let \(W\) be the Weyl group of this root system. We classify all simple modules over the twisted group ring \(\mathcal H[W]\) which belong to \(\mathcal M(\mathcal H,\mathcal A)\) as \(\mathcal H\)-modules. In §4 we show that the twisted group ring \(\mathcal H[W]\) is Morita equivalent to \(\mathcal HW\), the ring of \(W\)-invariants.
In §5 we establish a bijection between the set of simple modules over the algebra \(\mathcal H[W]\) associated with a semisimple simply-connected group \(G\), and the set of pairs \((P, \alpha)\), where \(P\) is a semistable principal \(G\)-bundle on the elliptic curve \(E = \mathbb C^*/q^\mathbb Z\), and \(\alpha\) is a certain “admissible representation” (cf. Definition 5.4) of the finite group \(\operatorname{Aut}(P)/(\operatorname{Aut} P)^\circ\). Our bijection is constructed by combining the results of §3 with a bijection between \(q\)-conjugacy classes in a loop group and \(G\)-bundles the elliptic curve \(E\), established earlier by some of us in [Baranovsky and Ginzburg, Int. Math. Res. Not. 1996, No. 15, 733–751 (1996; Zbl 0992.20034)].
For the entire collection see [Zbl 1027.00015].

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
16T20 Ring-theoretic aspects of quantum groups
14H60 Vector bundles on curves and their moduli

Citations:

Zbl 0992.20034
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