Non-linear finite $W$-symmetries and applications in elementary systems

TL;DR

This paper explores non-linear finite W-symmetries, reviewing their theory, presenting new results, and discussing potential physical applications like gauge theories based on non-linear algebras.

Contribution

It provides a comprehensive review of finite W-algebras, introduces new results on their structure and representations, and discusses how to construct physical theories using non-linear symmetries.

Findings

Finite W coadjoint orbits characterized.

New real forms and unitary representations identified.

Poincare-Birkhoff-Witt theorems for finite W-algebras established.

Abstract

In this paper it is stressed that there is no {\em physical} reason for symmetries to be linear and that Lie group theory is therefore too restrictive. We illustrate this with some simple examples. Then we give a readable review on the theory finite $W$-algebras, which is an important class of non-linear symmetries. In particular, we discuss both the classical and quantum theory and elaborate on several aspects of their representation theory. Some new results are presented. These include finite $W$ coadjoint orbits, real forms and unitary representation of finite $W$-algebras and Poincare-Birkhoff-Witt theorems for finite $W$-algebras. Also we present some new finite $W$-algebras that are not related to $sl(2)$ embeddings. At the end of the paper we investigate how one could construct physical theories, for example gauge field theories, that are based on non-linear algebras.