Abstract
We review Poisson–Lie groups and their applications in gauge theory and integrable systems from a mathematical physics perspective. We also comment on recent results and developments and their applications. In particular, we discuss the role of quasitriangular Poisson–Lie groups and dynamical r-matrices in the description of moduli spaces of flat connections and the Chern–Simons gauge theory.
Highlights
Phase spaces of physical systems are usually Poisson manifolds, and Lie group actions on these manifolds arise from physical or gauge symmetries of these systems
We show that Lie bialgebras are the infinitesimal structures associated with Poisson–Lie groups
To relate Poisson–Lie groups and Lie bialgebras, we describe the Poisson bracket of a Poisson–Lie group G in terms of a Poisson bivector BG ∈ Λ2 ( TG ) that assigns to every point g ∈ G an antisymmetric element B( g) ∈ Tg G ⊗ Tg G
Summary
Academic Editors: Ángel Ballesteros, Giulia Gubitosi and Francisco J. It is meant to provide an introduction that is accessible to physicists and mathematicians with no background on this topic, while at the same time covering current results. Due to the large body of work on Poisson–Lie groups and their numerous applications, it is impossible to do justice to all the work on this topic. For this reason, we focus on certain aspects and omit many others. Short and accessible introductions to the topic with emphasis on different aspects are given in lecture notes by A. A good introduction to Poisson geometry and Poisson–Lie groups is the textbook [4]. For an accessible introduction to dynamical r-matrices, see the textbook [8]
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