Introduction to finite $W$-algebras

TL;DR

This paper provides an introductory overview of finite W-algebras, covering their definition, motivation, structure, and representations, based on lecture notes that trace the historical development of the subject.

Contribution

It offers a comprehensive introduction to finite W-algebras, including their foundational concepts, motivations, and key structural and representational properties.

Findings

Defines infinite and finite W-algebras

Explains the motivation behind finite W-algebras

Summarizes basic structural and representation theory results

Abstract

These are notes of lectures given at UN Encuentro 2016 at the Colombia National University. We begin with the definition of infinite $W$-algebras. Then we explain the motivation for the definition if finite $W$-algebras. Then we present basic facts about the structure and representations of finite $W$-algebras. In these lectures we follow the historical development of the subject.