Affine and cyclotomic webs

TL;DR

This paper introduces the affine web category, a diagrammatic monoidal category generalizing polynomial web categories, and explores its connections to cyclotomic web categories and finite W-algebras, providing new algebraic presentations.

Contribution

It defines the affine web category over any commutative ring, constructs integral bases, and links cyclotomic web categories to W-algebras, advancing diagrammatic algebraic frameworks.

Findings

Constructed integral bases for affine and cyclotomic web categories.

Established connections between cyclotomic web categories and finite W-algebras.

Provided a diagrammatic presentation of certain W-Schur algebra subalgebras.

Abstract

Generalizing the polynomial web category, we introduce a diagrammatic $\Bbbk$-linear monoidal category, the affine web category, for any commutative ring $\Bbbk$. Integral bases consisting of elementary diagrams are obtained for the affine web category and its cyclotomic quotient categories. Connections between cyclotomic web categories and finite $W$-algebras are established, leading to a diagrammatic presentation of idempotent subalgebras of $W$-Schur algebras introduced by Brundan-Kleshchev. The affine web category will be used as a basic building block of another $\Bbbk$-linear monoidal category, the affine Schur category, formulated in a sequel.