Mixmaster model is associated to Borcherds algebra

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.