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4.5 The affine case

The affine case is characterized by the conditions that the Cartan matrix has vanishing determinant, is symmetrizable and is such that its symmetrization S is positive semi-definite (only one zero eigenvalue). As before, we also take the Cartan matrix to be indecomposable. By a reasoning analogous to what we did in Section 3.4 above, one can show that the radical of S is one-dimensional and that the ranks of S and A are equal to n − 1.

One defines the corresponding Kac–Moody algebras in terms of 3n + 1 generators, which are the same generators hi,ei,fi subject to the same conditions (4.10View Equation, 4.11View Equation) as above, plus one extra generator η which can be taken to fulfill

[η,hi] = 0, [η,ei] = δ1ie1, [η,fi] = − δ1if1. (4.25 )
The algebra admits the same triangular decomposition as above,
𝔤 = 𝔫− ⊕ 𝔥 ⊕ 𝔫+, (4.26 )
but now the Cartan subalgebra 𝔥 has dimension n + 1 (it contains the extra generator η).

Because the matrix Aij has vanishing determinant, one can find ai such that ∑ aiAij = 0 i. The element c = ∑ a h i i i is in the center of the algebra. In fact, the center of the Kac–Moody algebra is one-dimensional and coincides with ℂc [116Jump To The Next Citation Point]. The derived algebra ′ 𝔤 = [𝔤, 𝔤] is the subalgebra generated by hi,ei,fi and has codimension one (it does not contain η). One has

𝔤 = 𝔤 ′ ⊕ ℂ η (4.27 )
(direct sum of vector spaces, not as algebras). The only proper ideals of the affine Kac–Moody algebra 𝔤 are 𝔤′ and ℂc.

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.


Table 11: Untwisted affine Kac–Moody algebras.

Name

Dynkin diagram

A+1

PIC

   

A+ (n > 1) n

PIC

   

B+ n

PIC

   

C+n

PIC

   

D+ n

PIC

   

+ G2

PIC

   

F+4

PIC

   

+ E6

PIC

   

+ E7

PIC

   

+ E8

PIC



Table 12: Twisted affine Kac–Moody algebras. We use the notation of Kac [116Jump To The Next Citation Point].

Name

Dynkin diagram

A(22)

PIC

   

(2) A2n (n ≥ 2 )

PIC

   

A(22n)−1 (n ≥ 3)

PIC

   

(2) Dn+1

PIC

   

E(2) 6

PIC

   

(3) D4

PIC



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