On a Hidden Supersymmetry of Cosmological Billiards

Abstract

In the Belinski-Khalatnikov-Lifshitz cosmological models with an oscillatory approximation to singularity, the tracks of cosmological billiards (whatever they are) are realized in the Weyl chambers of the hyperbolic Lie algebras. The latter were classified by Li Wang Lai; their super analogs - the almost affine Lie superalgebras - were classified in arXiv:0906.1860. Here, we observe that some of the 142 hyperbolic Lie algebras $H$ can be ''superized'' to almost affine Lie superalgebras $S(H)$: for each of these $H$, it is possible to divide a row of its Cartan matrix by 2, simultaneously changing the parity of the corresponding Chevalley generators and relations between them. For the $H$ with a symmetrizable Cartan matrix, we list all 97 such pairs ($H\longleftrightarrow S(H)$). Several (18 of the total 66) thus ''superizable'' algebras $H$ have multiple such ''superizations''. The tracks of cosmological billiards corresponding to both terms of these pairs ($H\longleftrightarrow S(H)$) coincide since $H$ and $S(H)$ have one and the same Weyl chamber. We also classify ''superizations'' of hyperbolic Lie algebras with non-symmetrizable Cartan matrix, and pairs $\mathfrak{g}\longleftrightarrow \mathfrak{g}_{\bar 0}$, where $\mathfrak{g}$ is a simple finite-dimensional Lie superalgebra without Cartan matrix and simple component $\mathfrak{g}_{\bar 0}$.