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

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

1995-03-23

arxiv.org/abs/hep-th/9503161v1

Phys.Rept.272:139-214,1996

Physics

High Energy Physics

High Energy Physics - Theory

88 pages, LaTeX

Scientific paper

10.1016/0370-1573(95)00075-5

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.

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