Mixmaster model is associated to Borcherds algebra

TL;DR

This paper explores the integrability of the mixmaster model, linking it to Borcherds algebra through generalized root systems and Cartan matrices, revealing deep algebraic structures underlying its dynamics.

Contribution

It establishes a novel connection between the mixmaster model and Borcherds algebra, extending the algebraic framework used to analyze cosmological billiard models.

Findings

The mixmaster model is formulated as a pseudo-Euclidean generalized Toda chain.

A generalized Cartan matrix is constructed using root vectors in Minkowski space.

The model's algebraic structure is identified as a Borcherds algebra, generalizing known hyperbolic Kac--Moody algebras.

Abstract

The problem of integrability of the mixmaster model as a dynamical system with finite degrees of freedom is investigated. The model belongs to the class of pseudo-Euclidean generalized Toda chains. It is presented as a quasi-homogeneous system after transformations of phase variables. An application of the method of getting of Kovalevskaya exponents to the model leads to the generalized Adler -- van Moerbeke formula on root vectors. A generalized Cartan matrix is constructed with use of simple root vectors in Minkowski space. The mixmaster model is associated to a Borcherds algebra. The known hyperbolic Kac -- Moody algebra of Chitre billiard model is obtained by using three spacelike (without isotropic) root vectors.