Gröbner bases for fusion products

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 Gröbner 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.

18 pages