- Title: Handlebody diagram algebras
- Authors: Daniel Tubbenhauer and Pedro Vaz
- Status: Preprint. Last update: Tue, 18 May 2021 17:46:39 UTC
- ArXiv link: https://arxiv.org/abs/2105.07049
In this paper we study handlebody versions of classical diagram algebras, most prominently, handlebody versions of Temperley-Lieb, blob, Brauer/BMW, Hecke and Ariki-Koike algebras. Moreover, motivated by Green-Kazhdan-Lusztig's theory of cells, we reformulate the notion of (sandwich, inflated or affine) cellular algebras. We explain this reformulation and how all of the above algebras are part of this theory.
A few extra words
Our starting point is a diagrammatic description of handlebody braid groups of genus $g$, i.e. a diagrammatic description of the configuration space of a disk with $g$ punctures. The pictures hereby are e.g.
Handlebody Temperley--Lieb and blob algebras. The pictures to keep in mind are crossingless matchings
and core strands
respectively crossingless matchings decorated with colored blobs
Handlebody Brauer and BMW algebras. These are tangle algebras with core strands
and the picture is:
Handlebody Hecke and Ariki-Koike algebras, where the pictures are: