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

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.

88 pages, LaTeX

References in corpus (13)

• Two Dimensional Gauge Theories Revisited
• W-symmetry in Conformal Field Theory
• Poisson Structure Induced (Topological) Field Theories
• Representation theory of finite W algebras
• Covariant $W$ Gravity \& its Moduli Space from Gauge Theory
• W Gravity From Chern--Simons Theory
• Extensions of the Virasoro Algebra and Gauged WZW Models
• A2 Toda theory in reduced WZNW framework and the representations of the W algebra
• A Class of W-Algebras with Infinitely Generated Classical Limit
• Finite W-algebras
• The Covariant W_3 Action
• The Global Phase Space Structure of the Wess-Zumino-Witten Model
• The Effective Action of $W_3$ Gravity to All Orders

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