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

TL;DR

The paper broadens the scope of symmetry beyond linear Lie algebras by developing a thorough treatment of finite W-algebras, both classically via Kirillov structures and Poisson reduction and quantum mechanically through BRST quantization. It provides numerous concrete constructions and examples (notably W3^{(2)}) arising from sl2 embeddings and beyond, and develops the Miura transformation as a bridge to simpler subalgebras, together with a detailed analysis of coadjoint orbits and representation theory, including real forms and unitarity. It also surveys potential physical applications, particularly in one-dimensional gauge-like systems and Toda-type models, while outlining significant open problems for higher dimensions and non-sl2-based constructions. Overall, the work establishes finite W-algebras as rich, physically relevant non-linear symmetry structures with a robust mathematical framework and a range of possible physical realizations.

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.