One defines the corresponding Kac–Moody algebras in terms of generators, which are the same
generators
subject to the same conditions (4.10
, 4.11
) as above, plus one extra generator
which can be taken to fulfill
Because the matrix has vanishing determinant, one can find
such that
. The
element
is in the center of the algebra. In fact, the center of the Kac–Moody
algebra is one-dimensional and coincides with
[116
]. The derived algebra
is the
subalgebra generated by
and has codimension one (it does not contain
). One has
Affine Kac–Moody algebras appear in the BKL context as subalgebras of the relevant Lorentzian Kac–Moody algebras. Their complete list is known and is given in Tables 11 and 12.
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