Mixmaster model associated to a Borcherds algebra

Abstract

The problem of integrability of the mixmaster model as a dynamical system with finite degrees of freedom is studied. 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. Application of the method of getting Kovalevskaya exponents to the model leads to the generalized Adler–van Moerbeke formula for root vectors. A generalized Cartan matrix is constructed using simple root vectors inMinkowski space. The mixmaster model is associated to a Borcherds algebra. The known hyperbolic Kac–Moody algebra of the Chitre´ billiard model is obtained by using three spacelike root vectors.