On the geometric realization of the inner product and canonical basis for quantum affine $\mathfrak{sl}_n$

TL;DR

This paper provides a geometric interpretation of the inner product on the quantum group of affine rak{}n, leading to positivity results and a new geometric construction of the canonical basis.

Contribution

It introduces a novel geometric interpretation of the inner product and canonical basis for quantum affine rak{}n, with applications to positivity and basis construction.

Findings

Positivity of the inner product established

New geometric construction of the canonical basis

Enhanced understanding of quantum affine rak{}n structures

Abstract

We give a geometric interpretation of the inner product on the modified quantum group of $\hat{\mathfrak{sl}}_n$. We also give some applications of this interpretation, including a positivity result for the inner product, and a new geometric construction of the canonical basis.