Webs, Symmetrizers, and the Oriented Skein Category

Abstract

We define and study the two-colour 𝖌𝖑-web categories Web↑(𝖌𝖑) and Web↑↓(𝖌𝖑), which generalize the 𝖌𝖑-web categories that have previously appeared in the literature. After establishing some basic properties of these categories, we show that Web↑↓(𝖌𝖑) is isomorphic to a partial idempotent completion of the oriented skein category 𝒪𝒮. This completion is obtained from 𝒪𝒮 by adjoining objects corresponding to the quantum symmetrizer and antisymmetrizer morphisms. We use this isomorphism to generalize results about web categories from the literature. In particular, we study the web incarnation functors from Web↑↓(gl) to the category of type 1 weight modules for the quantum enveloping superalgebra 𝑈_q(𝖌𝖑_m|n). Our approach allows us to translate fullness and kernel results for 𝒪𝒮-incarnation functors into corresponding theorems for web incarnation functors, extending existing results to new cases where the base ring is not a field.