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
The algebra admits the same triangular decomposition as above, but now the Cartan subalgebra has dimension (it contains the extra generator ).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
(direct sum of vector spaces, not as algebras). The only proper ideals of the affine Kac–Moody algebra are and .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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