Project Details
Projekt
The (colored) sln link homologies and the categorical branching rules: Two methods combined
Applicant
Dr. Daniel Tubbenhauer
Subject Area
Mathematics
Term
from 2014 to 2015
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 269493184
It follows from my newest paper that one can obtain the (colored) sln link homologies purely combinatorial using the so-called cyclotomic KL-R algebra Rm. The chain groups are Rm-modules and the differentials are Rm-module homomorphisms in my formulation.This algebra is related to slm (with m in general not equal n) and has an action of Sn (the symmetric group in n-1 generators).This purely combinatorial formulation is surprising, since it is known that the sln homologies are highly connected to topology and geometry. For example, the are closely related to knot Floer homology - which comes from differential geometry.The branching rule is a very neat ''tool'' from the classical representation theory of Lie algebras and symmetric groups. It works roughly as follows: Consider the embedding of sl(k-1) into slk. This induces the so-called restriction functor res from slk-mod to sl(k-1)-mod. One can decompose a slk-module as a direct sum of sl(k-1)-modules (where sl(k-1) is simpler as an algebra and has simpler modules). This procedure can be used recursively until one hits a direct sum of ''trivial'' sl1-modules. Roughly the same works for Sn.On the other hand, one can also build the starting module inductively using combinatorial steps from simple building blocks.Pedro Vaz (host) has ''categorified'' this classical representation theoretical method: He describes in one of his newest papers the branching rule for the cyclotomic KL-R algebra Rm coming from them embedding of sl(m-1) into slm. One can, as a consequence, decompose a Rm-module as a direct sum of R(m-1)-modules. One can again use this procedure recursively until one has decomposed the starting module as a direct sum of ''trivial'' R1-modules. On the other hand, one can also build the starting module combinatorial step-by-step from simple building blocks.The idea for the project is to combine both methods: It follows from my work that the sln link homologies can be described using Rm-modules and Rm-module homomorphisms and it follows from Vaz's work that one can decompose Rm-modules into simple pieces and build them from simple building blocks.Thus, the question we would like to answer is if we can decompose the sln homologies into simpler pieces and, on the other hand, build them from simple building blocks. We note that this question (how the sln homology is connected to the sl(n-1) homology) is an open question since the very beginnings of Khovanov link homologies.Moreover, except in the case n=2, n=3, literally nothing is known about the sln homologies. Thus, a related question is if we can use the new interpretation to deduce new results about the higher homologies or if we maybe obtain new computation methods for these homologies.Furthermore, there is another possible branching rule for these homologies: The one that comes from the action of Sn mentioned above. Note that this would reveal the purely combinatorial structure of these sln link homologies.
DFG Programme
Research Fellowships
International Connection
Belgium
