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Abstract

The involutory subalgebra K(E$_9$) of the affine Kac-Moody algebra E$_9$ was
recently shown to admit an infinite sequence of unfaithful representations of
ever increasing dimensions arXiv:2102.00870. We revisit these representations
and describe their associated ideals in more detail, with particular emphasis
on two chiral versions that can be constructed for each such representation.
For every such unfaithful representation we show that the action of K(E$_9$)
decomposes into a direct sum of two mutually commuting (`chiral' and
`anti-chiral') parabolic algebras with Levi subalgebra
$\mathfrak{so}(16)_+\,\oplus\,\mathfrak{so}(16)_-$. We also spell out the
consistency conditions for uplifting such representations to unfaithful
representations of K(E$_{10}$). From these results it is evident that the
holonomy groups so far discussed in the literature are mere shadows (in a
Platonic sense) of a much larger structure.