Finite Crystals and Paths

TL;DR

This paper explores the structure of finite crystals in quantum affine algebras, establishing an isomorphism with path sets and demonstrating their relation to tensor products and Bethe Ansatz solutions.

Contribution

It introduces a new perspective on finite crystals, including non-perfect objects, and links path models to tensor products and Bethe Ansatz in quantum affine algebras.

Findings

Set of paths is isomorphic to a direct sum of crystals.

Examples show the direct sum can form a tensor product.

Connections to Bethe Ansatz are demonstrated.

Abstract

We consider a category of finite crystals of a quantum affine algebra whose objects are not necessarily perfect, and set of paths, semi-infinite tensor product of an object of this category with a certain boundary condition. It is shown that the set of paths is isomorphic to a direct sum of infinitely many, in general, crystals of integrable highest weight modules. We present examples from C_n^{(1)} and A_{n-1}^{(1)}, in which the direct sum becomes a tensor product as suggested from the Bethe Ansatz.