Superalgebra deformations of web categories: Affine and cyclotomic webs

TL;DR

This paper introduces a new diagrammatic supercategory called $ extbf{Web}^{aff}_A$, which generalizes and deforms existing web categories using superalgebra structures, and establishes basis results and functorial connections to module categories.

Contribution

It defines the affine and cyclotomic web categories associated with Frobenius superalgebras, extending previous web categories and providing basis and functoriality results.

Findings

Established a basis of decorated double coset diagrams for morphism spaces.

Constructed asymptotically faithful functors to endofunctor categories of $ ext{gl}_n(A)$-modules.

Defined and analyzed cyclotomic quotient categories with basis results.

Abstract

Let $\mathbb{k}$ be a characteristic zero domain. We define and study a diagrammatic monoidal $\mathbb{k}$-linear supercategory $\mathbf{Web}^{aff}_{A}$ associated to any locally unital Frobenius $\mathbb{k}$-superalgebra $A$. This category can be viewed variously as an affinization of the finite web category $\mathbf{Web}_{A}$ previously defined by the authors and Zhu, as a thickening of the degenerate affine wreath product algebras defined by Savage, or as a Frobenius deformation of affine web categories defined by Song and Wang. We show that there is an asymptotically faithful family of functors from $\mathbf{Web}^{aff}_{A}$ to the monoidal supercategory of endofunctors of $\mathfrak{gl}_n(A)$-modules for every $n \geq 1$, and use this to establish a basis of `decorated double coset diagrams' for morphism spaces in $\mathbf{Web}^{aff}_{A}$. We also define and establish basis results for the cyclotomic quotient category $\mathbf{Web}^{\Lambda}_{A}$ associated with a cyclotomic datum $\Lambda$.