On the Hall algebra of an elliptic curve, I

By I. Burban, O. Schiffmann. 2005

212 citations recorded by the source corpus.

This paper is a sequel to math.AG/0505148, where the Hall algebra U^+_E of the category of coherent sheaves on an elliptic curve E defined over a finite field was explicitly described, and shown to be a two-parameter deformation of the ring of diagonal invariants R^+=C[x_1^{\pm 1}, ..., y_1,...]^{S_{\infty}} (in infinitely many variables). In the present work, we study a geometric version of this Hall algebra, by considering a convolution algebra of perverse sheaves on the moduli spaces of coherent sheaves on E. This allows us to define a "canonical basis" B of U^+_E in terms of intersection cohomology complexes. We also give a characterization of this basis in terms of an involution, a lattice and a certain PBW-type basis. Finally we provide some (loose) interpretation of the above construction in terms of Schubert varieties in (nonexistent!) "double affine grassmanian", and in terms of character sheaves of loop groups (both of type A).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1cf73df09a4f2c493bd71dae9ac7c2edd50fa10f

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In Geometry and Topology

1. François Bergeron
2. Andrei Neguct
3. E. Gorsky
4. Adriano M. Garsia
5. Emily Sergel Leven
6. Guoce Xin
7. I. Gordon
8. Andrei Neguț

In Mathematical Physics

1. Andrei Neguct
2. E. Gorsky
3. François Bergeron
4. I. Gordon
5. Adriano M. Garsia
6. Emily Sergel Leven
7. Guoce Xin
8. Andrei Neguț

Citations in the corpus

• The Shuffle Algebra Revisited (2012, methodology): “The elliptic Hall algebra E was studied in detail by Burban and Schiffmann in [1], and we would like to compare the shuffle algebra with their viewpoint.”
• The affine Yangian of gl1 revisited (2014, unknown): “We recall the elliptic Hall algebra studied in [2].”
• Compositional (km,kn)-Shuffle Conjectures (2014, background): “Its connection to the algebraic geometrical developments is that this algebra is a concrete realization of a portion of the Elliptic Hall Algebra studied by Schiffmann and Vasserot in [21], [22], and [23].”
• Refined knot invariants and Hilbert schemes (2013, methodology): “These commutation relations were discovered by Burban and Schiffmann in [6].”
• Quantum groups via Hall algebras of complexes (2011, unknown): “Let D(A) denote the (Z-graded) bounded derived category of A, with its shift functor [1].”
• Quantum Geometry and Quiver Gauge Theories (2013, unknown): “It was called quantum continuous gl∞, or deformation of universal enveloping of qdifference operators in one-variable in [177, 178]; spherical elliptic Hall algebra or gl∞ DAHA in [179] and [165, 180].”
• Symplectic reflection algebras (2007, background): “ Hopf algebras, [4]; real algebraic geometry, [75]; symmetricfunction theory, [9]; and mostofall doubleaffineHeckealgebras,[30] andthe references in it and [134], (with a new relation to Hall algebras– [29] and [128]). To try compensate for these omissions the bibliography includes references to all t”