  We study the quantum dynamics of a supersymmetric squashed three-sphere by
dimensionally reducing (to one timelike dimension) the action of D=4 simple
supergravity for an SO(3)-homogeneous (Bianchi IX) cosmological model. The
quantization of the homogeneous gravitino field leads to a 64-dimensional
fermionic Hilbert space. The algebra of the supersymmetry constraints and of
the Hamiltonian one is found to close. One finds that the quantum Hamiltonian
is built from operators that generate a 64-dimensional representation of the
(infinite-dimensional) maximally compact sub-algebra of the rank-3 hyperbolic
Kac-Moody algebra AE_3. Some exponentials of these operators generate a
spinorial extension of the Weyl group of AE_3 which describe (in the small
wavelength limit) the chaotic quantum evolution of the universe near the
cosmological singularity.
