Hidden symmetries in mixmaster-type universe

TL;DR

This paper investigates a multidimensional mixmaster universe model, revealing a hidden symmetry through algebraic methods involving Kac-Moody algebras, and finds it is non-chaotic due to infinite configuration space volume.

Contribution

It introduces an algebraic approach using Cartan matrices to identify hidden symmetries in a multidimensional mixmaster universe model, expanding understanding of its algebraic structure.

Findings

The model is associated with a simply-laced Lorentzian Kac-Moody algebra.

The algebra is not hyperbolic; the Weyl vector's square is negative.

The infinite volume of configuration space implies the model is not chaotic.

Abstract

A model of multidimensional mixmaster-type vacuum universe is considered. It belongs to a class of pseudo-Euclidean chains characterized by root vectors. An algebraic approach of our investigation is founded on construction of Cartan matrix of the spacelike root vectors in Wheeler -- DeWitt space. Kac -- Moody algebras can be classified according to their Cartan matrix. By this way a hidden symmetry of the model considered is revealed. It is known, that gravitational models which demonstrate chaotic behavior are associated with hyperbolic Kac -- Moody algebras. The algebra considered in our paper is not hyperbolic. The square of Weyl vector is negative. The mixmaster-type universe is associated with a simply-laced Lorentzian Kac -- Moody algebra. Since the volume of the configuration space is infinite, the model is not chaotic.