Modul:   MAT789  Quantum Topology Seminar

Beyond the nilHecke algebra: categorifying Kac-Moody algebras

Vortrag von Aaron Lauda

Sprecher eingeladen von: Prof. Dr. Anna Beliakova

Datum: 20.05.11  Zeit: 13.15 - 15.00  Raum: Y27H12

In this series of talks I will introduce and motivate the diagrammatic construction of categorified quantum groups. These structures were topologically motivated by the goal of categorifying quantum link invariants and quantum 3-manifold invariants. In the first talk (talks in Mathematical Physics, May 5) I will review the representation theory of the quantum group sl2. In categorified representation theory vector spaces are replaced by categories, and endomorphisms of vector spaces by functors between categories. In this setting one can look for higher structure in the form of natural transformations between functors. This higher structure could not be seen in traditional representation theory. This lecture will focus on a well known categorification of the irreducible representations using cohomology rings of Grassmannians and partial flag varieties. This motivating example reveals a great deal of the structure present in categorified quantum groups. The second talk will further probe this higher structure and explain the fundamental connection between the nilHecke algebra and quantum sl2. We will define a categorification of the quantum group associated to sl2. The third talk will explain how to generalize the nilHecke algebra to obtain categorifications of more general Lie algebras, namely symmetrizable Kac-Moody algebras. No prior knowledge of Kac-Moody algebras will be assumed. These categorifications are all obtained from a planar diagrammatic calculus.