Alistair Savage (U. Ottawa) : A graphical categorification of the Heisenberg algebra
In this talk, we will present a graphical category in terms of certain planar braid-like diagrams. The definition of this category is inspired by the representation theory of Hecke algebras of type A (which are certain deformations of the group algebra of the symmetric group). The Heisenberg algebra (in infinitely many generators), which plays an important role in the description of certain quantum mechanical systems, injects into the Grothendieck group of our category, yielding a "categorification" of this algebra. We will also see that our graphical category acts on the category of modules of Hecke algebras and of general linear groups over finite fields. Additionally, other algebraic structures, such as the affine Hecke algebra, appear naturally.
We will assume no prior knowledge of Hecke algebras or the Heisenberg algebra. This is joint work with Anthony Licata and inspired by work of Mikhail Khovanov.