Data

  • 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

Abstract

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.

This illustrates a handlebody braid of genus $4$: The three strands on the right are usual strands. The four thick and blue/grayish strands on the left are core strands and they correspond to the punctures of the disk respectively the cores of the handlebody. The point is that via an appropriate closure, {\ie} merging the core strands at infinity, illustrated by
The core strands correspond to cores of a handlebody, hence the name. There are certain handlebody algebra that we study in this paper, based on the above diagrammatic description:
  • Handlebody Temperley--Lieb and blob algebras. The pictures to keep in mind are crossingless matchings and core strands (left, Temperley--Lieb) respectively crossingless matchings decorated with colored blobs (right, blob):
  • 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:
For genus $g=0$ our algebras gives the well-known algebras of the same name, for genus $g=1$ our algebras recover (extended) affine versions of these algebras.