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On the Hall algebra of an elliptic curve. I. (English) Zbl 1286.16029

In this paper the authors give a description of the Hall algebra of an elliptic curve defined over a finite field and show that the group \(\mathrm{SL}(2,\mathbb Z)\) of exact autoequivalences of the bounded derived category of coherent sheaves on \(X\) acts on the Drinfeld double of the Hall algebra by algebra automorphisms. The authors also give a presentation, by generators and relations, of a subalgebra of the Hall algebra which is shown to coincide with Kapranov’s spherical Hall algebra and of its Drinfeld double. The latter Drinfeld double is shown to be isomorphic to a flat two-parameter deformation of the ring of symmetric polynomials in two sets of countably many variables under the simultaneous action of the infinite symmetric group.

MSC:

16T05 Hopf algebras and their applications
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14H52 Elliptic curves
17B37 Quantum groups (quantized enveloping algebras) and related deformations
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
16G20 Representations of quivers and partially ordered sets
16T20 Ring-theoretic aspects of quantum groups
18E30 Derived categories, triangulated categories (MSC2010)
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