Combinatorics of infinite rank module categories over finite dimensional $\mathfrak{sl}_3$-modules in Lie-algebraic context

TL;DR

This paper explores the combinatorial structure of transitive module categories over finite dimensional rak{sl}_3-modules, revealing eight graph families that generalize classical infinite Dynkin diagrams within a Lie-algebraic framework.

Contribution

It classifies the combinatorics of these module categories and introduces new rak{sl}_3-generalized infinite Dynkin diagrams.

Findings

Identified eight rak{sl}_3-related graph families

Connected module category combinatorics to classical Dynkin diagrams

Provided a framework for understanding rak{sl}_3-module actions

Abstract

We determine the combinatorics of transitive module categories over the monoidal category of finite dimensional $\mathfrak{sl}_3$-modules which arise when acting by the latter monoidal category on arbitrary simple $\mathfrak{sl}_3$-modules. This gives us a family of eight graphs which can be viewed as $\mathfrak{sl}_3$-generalizations of the classical infinite Dynkin diagrams.