Poisson algebra of 2d dimensionally reduced gravity

TL;DR

This paper develops a mathematical framework using Lax pairs and algebraic structures to analyze two-dimensional reduced vacuum Einstein's equations, revealing non-ultralocal Poisson brackets and modified Yang-Baxter equations.

Contribution

It introduces a novel r-matrix formulation based on twisted affine algebras for 2D gravity reduction, providing tools to compute classical observables with boundary conditions.

Findings

Derived a Poisson algebra with non-ultralocal brackets

Established a pure c-number modified Yang-Baxter equation

Outlined a method to obtain classical observables

Abstract

Using a Lax pair based on twisted affine $sl(2,R)$ Kac-Moody and Virasoro algebras, we deduce a r-matrix formulation of two dimensional reduced vacuum Einstein's equations. Whereas the fundamental Poisson brackets are non-ultralocal, they lead to pure c-number modified Yang-Baxter equations. We also describe how to obtain classical observables by imposing reasonable boundaries conditions.