An Introduction to Canonical Bases

摘要

The aim of this article is to give an introduction to canonical bases (= global crystal) of quantum universal enveloping algebras in a geometric language. In other words, we explain how to use the theory of perverse sheaves and Ringel-Hall algebras in the study of quantum universal enveloping algebras. Another purpose is to explain a combinatorial structure (which is called the crystal structure) on the canonical bases.