Gr\"obner bases for fusion products

TL;DR

This paper introduces a new approach using Gr"obner bases to analyze fusion products in the representation theory of current Lie algebras, providing a proof for the conjecture in the  case.

Contribution

It presents a novel method employing Gr"obner theory to study fusion products and outlines a strategy to prove related conjectures, successfully applied to  Lie algebra.

Findings

New Grner basis approach to fusion products

Proof of the conjecture for  case

Strategy applicable to other Lie algebras

Abstract

We provide a new approach towards the analysis of the fusion products defined by B.~Feigin and S.~Loktev in the representation theory of (truncated) current Lie algebras. We understand the fusion product as a degeneration using Gr\"obner theory of non-commutative algebras and outline a strategy on how to prove a conjecture about the defining relations for the fusion product of two evaluation modules. We conclude with following this strategy for $\mathfrak{sl}_2(\mathbb{C}[t]) $ and hence provide yet another proof for the conjecture in this case.