Supergravity Black Holes and Billiards and Liouville integrable structure of dual Borel algebras

TL;DR

This paper demonstrates that supergravity equations for black holes and cosmic billiards are Liouville integrable due to a universal Poisson structure on dual Borel algebras, enabling explicit solution algorithms and classification tools.

Contribution

It reveals the universal Liouville integrability mechanism for supergravity solutions and constructs explicit integrable algorithms and conserved quantities for symmetric space models.

Findings

Supergravity equations are Liouville integrable due to Borel algebra structures.

Explicit integration algorithms for symmetric spaces are derived.

A complete set of conserved involutive Hamiltonians is constructed.

Abstract

In this paper we show that the supergravity equations describing both cosmic billiards and a large class of black-holes are, generically, both Liouville integrable as a consequence of the same universal mechanism. This latter is provided by the Liouville integrable Poissonian structure existing on the dual Borel algebra B_N of the simple Lie algebra A_{N-1}. As a by product we derive the explicit integration algorithm associated with all symmetric spaces U/H^{*} relevant to the description of time-like and space-like p-branes. The most important consequence of our approach is the explicit construction of a complete set of conserved involutive hamiltonians h_{\alpha} that are responsible for integrability and provide a new tool to classify flows and orbits. We believe that these will prove a very important new tool in the analysis of supergravity black holes and billiards.