A characterisation of indecomposable web-modules over Khovanov-Kuperberg Algebras

TL;DR

This paper characterizes indecomposable web-modules over Khovanov-Kuperberg algebras, linking their indecomposability to the Kuperberg bracket and explicit foam constructions, advancing understanding of their algebraic structure.

Contribution

It provides a criterion for indecomposability of web-modules based on the Kuperberg bracket and constructs explicit idempotents using foam diagrams, a novel approach.

Findings

Indecomposability corresponds to derivation from the Kuperberg bracket.

Explicit foam-based idempotents are constructed for web-modules.

Red graphs encode foam data and help identify indecomposable modules.

Abstract

After shortly recalling the construction of the Khovanov-Kuperberg algebras, we give a characterisation of indecomposable web-modules. It says that a web-module is indecomposable if and only if one can deduce it directly from the Kuperberg bracket (via a Schur lemma argument). The proofs relies on the construction of idempotents given by explicit foams. These foams are encoded by combinatorial data called red graphs. The key point is to show that when, for a web $w$ the Schur lemma does not apply, one can find an appropriate red graph for $w$.