Elementary construction of canonical bases, foldings, and piecewise linear bijections

TL;DR

This paper presents an elementary method for constructing canonical bases of quantum groups of finite type using folding theory and piecewise linear parametrization, avoiding geometric and crystal basis theories.

Contribution

It introduces a new elementary approach to construct canonical bases without relying on Lusztig's geometric theory or Kashiwara's crystal bases.

Findings

Constructed canonical bases via folding theory.

Developed a piecewise linear parametrization method.

Provided an elementary construction approach.

Abstract

Let ${\mathbf U}_q^-$ be the negative half of a quantum group of finite type. We construct the canonical basis of ${\mathbf U}_q^-$ by applying the folding theory of quantum groups, and piecewise linear parametrization of canonical basis. Our construction is elementary, in the sense that we don't appeal to Lusztig's geometric theory of canonical bases, nor to Kashiwara's theory of crystal bases.