Gr\"obner basis for fusion products
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 Grobner 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.