On a hidden supersymmetry of cosmological billiards

TL;DR

This paper classifies hyperbolic Lie algebras and their superizations, revealing a hidden supersymmetry in cosmological billiards where Weyl chambers are scale-related, and explores superizations beyond symmetrizable cases.

Contribution

It provides a comprehensive classification of hyperbolic Lie algebras and their superizations, uncovering a hidden supersymmetry in cosmological billiards.

Findings

97 pairs of hyperbolic Lie algebras and superized superalgebras identified

Multiple superizations found for some hyperbolic Lie algebras

Weyl chambers of pairs differ only by a scale, linking cosmological billiard tracks.

Abstract

The hyperbolic Lie algebras with symmetrizable Cartan matrix are classified, there are 142 of them some of which can be ``superized'' to an almost affine Lie superalgebra. We list all 97 pairs (a hyperbolic Lie algebra $H$, its superized almost affine Lie superalgebra $S(H)$). Several (18 of the total 66) superizable hyperbolic Lie algebras have multiple superizations. The tracks of cosmological billiards corresponding to both terms of these pairs ($H\leftrightarrow S(H)$) are the same since the Weyl chambers of $H$ and of $S(H)$ differ only by a~scale. We also classified ``superizations'' of hyperbolic Lie algebras with non-symmetrizable Cartan matrix. No interpretations of these Lie (super)algebras are known to us.