Commutative algebras in Drinfeld categories of abelian Lie algebras

TL;DR

This paper classifies and describes commutative algebras with non-degenerate forms in braided categories related to abelian Lie algebras, including their local modules and finite simple module classifications.

Contribution

It introduces a detailed framework for understanding commutative algebras in Drinfeld categories of abelian Lie algebras, including classification results.

Findings

Classification of commutative algebras with finite simple local modules

Description of local modules over these algebras

Framework for non-degenerate multiplicative forms in braided categories

Abstract

We describe (braided-)commutative algebras with non-degenerate multiplicative form in certain braided monoidal categories, corresponding to abelian metric Lie algebras (so-called Drinfeld categories). We also describe local modules over these algebras and classify commutative algebras with finite number of simple local modules.