Seminar by Daniel Tubbenhauer (Aarhus University)

Title: $\mathfrak{sl}_n$-link homologies using $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$-highest weight theory

2014.04.16 | Christine Dilling

Date Wed 23 Apr
Time 14:15 15:15
Location Aud D.3


The Jones polynomial is a celebrated invariant of links. One way to ``explain'' its appearance is the usage of the representation theory of $\textbf{U}_q(\mathfrak{sl}_2)$. Hence the name $\mathfrak{sl}_2$-link polynomial.

It has $\mathfrak{sl}_n$ variants using the representation theory of $\textbf{U}_q(\mathfrak{sl}_n)$ instead. Moreover, all of them have categorifications called the Khovanov-Rozansky $\mathfrak{sl}_n$-link homologies. These are certain chain complexes of graded vector spaces whose graded Euler characteristic gives the $\mathfrak{sl}_n$-link polynomials.

All of them have an underlying combinatorial and diagrammatic structure called $\mathfrak{sl}_n$-webs for the polynomials and $\mathfrak{sl}_n$-foams for the homologies. We will explain these in detail \textbf{focussing on $n=2$ and the level of webs} (this is just convenient: Everything works for any $n>1$ and on the level of the $\mathfrak{sl}_n$-link homologies).

Then we explain how the $\mathfrak{sl}_n$-webs form an irreducible $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$-module of a certain highest weight. This gives rise to an ``explanation'' of the $\mathfrak{sl}_n$-link polynomials using $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$-highest weight theory - in contrast to the Reshetikhin-Turaev approach that ``explains'' them using $\textbf{U}_q(\mathfrak{sl}_n)$-intertwiners.

The same works for the $\mathfrak{sl}_n$-link homologies: They can be ``explained'' using ``categorified'' highest weight theory of $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$: Khovanov-Lauda's categorification $\mathcal U(\mathfrak{sl}_m)$ of $\dot{\textbf{U}}_q(\mathfrak{sl}_m)$.

In fact, something better is going on that we do not explain in much detail: Since $\textbf{U}_q^-(\mathfrak{sl}_m)$ suffices on the level of $\mathfrak{sl}_n$-webs, the cyclotomic KL-R algebras suffices on the ``higher'' level.