CU Algebraic Lie Theory Seminar

CU Algebraic Lie Theory Seminar
Why representation theorists already know about quantum knot invariants Aaron Lauda, USC
November 27, 2012 Math 350 2-3pm	Abstract It is a well understood story that one can extract link invariants from quantum deformations of the universal enveloping algebra of simple Lie algebra called quantum groups. These invariants are called Reshetikhin-Turaev invariants and the famous Jones polynomial is the simplest example. Kauffman showed that the Jones polynomial could be described very simply by replacing crossings in a knot diagrams by various smoothings. In this talk we will explain Cautis-Kamnitzer-Licata's simple new approach to understanding these invariants using basic representation theory and the quantum Weyl group action based on the representation theoretic notion called skew-Howe duality. Even the graphical (or skein theory) description of these invariants can be recovered in an elementary way from this data. The advantage of this approach is that it suggests a `categorification' where knot homology theories arise in an elementary way from the structure of categorified quantum groups.

Why representation theorists already know about quantum knot invariants

Aaron Lauda, USC

November 27, 2012

Math 350

2-3pm

Abstract

It is a well understood story that one can extract link invariants from quantum deformations of the universal enveloping algebra of simple Lie algebra called quantum groups. These invariants are called Reshetikhin-Turaev invariants and the famous Jones polynomial is the simplest example. Kauffman showed that the Jones polynomial could be described very simply by replacing crossings in a knot diagrams by various smoothings. In this talk we will explain Cautis-Kamnitzer-Licata's simple new approach to understanding these invariants using basic representation theory and the quantum Weyl group action based on the representation theoretic notion called skew-Howe duality. Even the graphical (or skein theory) description of these invariants can be recovered in an elementary way from this data. The advantage of this approach is that it suggests a `categorification' where knot homology theories arise in an elementary way from the structure of categorified quantum groups.