### Representation Theory and Automorphic Forms, University of Vienna

**Peter Samuelson (UC Riverside)**

*Hall algebras and skein algebras*

Organizer's time: 2021-05-04 17:00 Europe/Vienna

Duration: 1 hour 15 minutes

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The Hall algebra "counts extensions" in an abelian category, and for categories of quiver representations this was related to quantum groups by Ringel and Green. The "elliptic Hall algebra" of Burban and Schiffmann is the Hall algebra of the category of sheaves over an elliptic curve, and this has since been related to many other objects (Hilbert schemes, knot homology, much more...). The skein algebra of a surface is the quotient of the space of links in the thickened surface by local "skein relations," which come from quantum groups and knot theory. In this talk we review these ideas and show that the elliptic Hall algebra is isomorphic to the "type A" skein algebra of the torus. We also briefly describe the "type BCD" skein algebra of the torus, and ask whether there is a Hall-theoretic interpretation. (This is based on joint works with Morton and Pokorny.)

Submitted by: Anton Mellit