Elementary construction of Lusztig's canonical basis

TL;DR

This paper provides an elementary, expository construction of Lusztig's canonical basis for type ADE using braid group techniques, highlighting its fundamental properties and applications.

Contribution

It offers a simplified, rank-two based approach to construct Lusztig's canonical basis, making its properties more accessible.

Findings

Canonical basis constructed explicitly for type ADE

Basis descends to all highest weight integrable representations

Basis exhibits crystal structure

Abstract

In this largely expository article we present an elementary construction of Lusztig's canonical basis in type ADE. The method, which is essentially Lusztig's original approach, is to use the braid group to reduce to rank two calculations. Some of the wonderful properties of the canonical basis are already visible; that it descends to a basis for every highest weight integrable representation, and that it is a crystal basis.