Handlebody diagram algebras

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. Contents1. Introduction 1 2. A generalization of cellularity 4 3. Handlebody braid and Coxeter groups 11 4. Handlebody Temperley-Lieb and blob algebras 16 5. Handlebody Brauer and BMW algebras 28 6. Handlebody Hecke and Ariki-Koike algebras 36 References 41

Cellular subalgebras of the partition algebra

“…It is expected that the wreath product construction will extend to generalizations of cellular algebras, such as affine cellular algebras [KX12], the recently defined skew graded cellular algebras [HMR21]. It should also work by replacing the underlying diagram algebra; for example, the base algebra could use the blob algebra B n (β, γ, δ), generalizations of the Temperley-Lieb algebra (see, e.g., [BCF22] and references therein), the higher genus diagram algebras [TV21], the BMW algebra [BW89,Mur87] or the associated tangle algebra [FG95], or quiver Hecke algebras (also known as Khovanov-Lauda-Rouquier (KLR) algebras [KL09,Rou08]) [HM10].…”

“…In turn, this is connected with another diagrammatic algebra known as the Khovanov-Lauda-Rouquier (KLR) algebra or quiver Hecke algebra [KL09,Rou08] that has a vast literature too large to even begin to list here, but we simply mention the generalized Schur-Weyl duality functors of Kang, Kashiwara, and Kim [KKK15,KKK18] (see [KKOP20,KKOP21] for some recent results related to these functors). A broader framework of sandwiched cellular algebras was introduced by Tubbenhauer and coauthors [MT21,Tub22,TV21] as a way to generalize Kazhdan-Lusztig cells to general algebras.…”