Handlebody diagram algebras

TL;DR

This paper introduces handlebody variants of classical diagram algebras, reformulates cellular algebra theory, and demonstrates how these algebras fit into this new framework, enriching the algebraic landscape.

Contribution

It develops handlebody versions of key diagram algebras and reformulates cellular algebra theory to include these new structures, expanding the scope of algebraic frameworks.

Findings

Handlebody versions of Temperley-Lieb, blob, Brauer, BMW, Hecke, and Ariki-Koike algebras are constructed.

A new reformulation of cellular algebra theory is proposed, encompassing these handlebody algebras.

The paper shows how these algebras fit into the generalized cellular algebra framework.

Abstract

In this paper we study handlebody versions of some 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.