Quadratic brackets from symplectic forms

TL;DR

This paper surveys Poisson-Lie symmetries in classical systems, highlighting the emergence of quadratic Poisson brackets in group-like variables and their potential link to quadratic exchange algebras after quantization.

Contribution

It provides a physicist-oriented overview of quadratic Poisson brackets in symplectic forms, with examples from geometric actions and the WZNW model, emphasizing their significance in classical and quantum contexts.

Findings

Quadratic Poisson brackets appear for group-like variables.

These brackets are linked to quadratic exchange algebras upon quantization.

The paper offers a unifying perspective on Poisson-Lie symmetries in classical systems.

Abstract

We give a physicist oriented survey of Poisson-Lie symmetries of classical systems. We consider finite dimensional geometric actions and the chiral WZNW model as examples for the general construction. An essential point is that quadratic Poisson bracets appear for group--like variables. It is believed that after quantization they lead to quadratic exchange algebras.