Verlinde rings and cluster algebras arising from quantum affine algebras

TL;DR

This paper explores a conjecture linking Verlinde rings and cluster algebras derived from quantum affine algebras, providing proofs in specific cases and evidence supporting the positivity conjecture.

Contribution

It formulates a positivity conjecture connecting Verlinde rings and cluster algebras, and proves it for certain Lie algebra types and levels, advancing understanding of their algebraic structures.

Findings

Proved the conjecture for simply-laced Lie algebras at level 2.

Established the conjecture for type A1 at arbitrary levels.

Showed all objects have positive quantum dimensions under certain conditions.

Abstract

We formulate a positivity conjecture relating the Verlinde ring associated with an untwisted affine Lie algebra at a positive integer level and a subcategory of finite-dimensional representations over the corresponding quantum affine algebra with a cluster algebra structure. Specifically, we consider a ring homomorphism from the Grothendieck ring of this representation category to the Verlinde ring and conjecture that every object in the category has a positive image under this map. We prove this conjecture in certain cases where the underlying simple Lie algebra is simply-laced with level 2 or of type $A_1$ at an arbitrary level. The proof employs the close connection between this category and cluster algebras of finite cluster type. As further evidence for the conjecture, we show that for any level, all objects have positive quantum dimensions under the assumption that some Kirillov-Reshetikhin modules have positive quantum dimensions.