100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

58 H. Nicolai4. Basics of Kac Moody TheoryWe here summarize some basic results from the theory of KM algebras,referring the reader to 15,16,17 for comprehensive treatments. Every KM algebrag ≡ g(A) can be defined by means of an integer-valued Cartan matrixA and a set of generators and relations. We shall assume that the Cartanmatrix is symmetrizable since this is the case encountered for cosmologicalbilliards. The Cartan matrix can then be written as (i, j = 1, . . . r, with rdenoting the rank of g(A))A ij = 2〈α i|α j 〉〈α i |α i 〉(43)where {α i } is a set of r simple roots, and where the angular brackets denotethe invariant symmetric bilinear form of g(A). 15 Recall that the rootscan be abstractly defined as linear forms on the Cartan subalgebra (CSA)h ⊂ g(A). The generators, which are also referred to as Chevalley-Serregenerators, consist of triples {h i , e i , f i } with i = 1, . . . , r, and for each iform an sl(2, R) subalgebra. The CSA h is then spanned by the elementsh i , so that[h i , h j ] = 0 (44)The remaining relations generalize the ones we already encountered in Eqs.(26) and (29): Furthermore,and[e i , f j ] = δ ij h j (45)[h i , e j ] = A ij e j , [h i , f j ] = −A ij f j (46)so that the value of the linear form α j , corresponding to the raising operatore j , on the element h i of the preferred basis {h i } of h is α j (h i ) = A ij . Moreabstractly, and independently of the choice of any basis in the CSA, theroots appear as eigenvalues of the adjoint action of any element h of theCSA on the raising (e i ) or lowering (f i ) generators: [h, e i ] = +α i (h)e i ,[h, f i ] = −α i (h)f i . Last but not least we have the so-called Serre relationsad (e i ) 1−Aij ( e j)= 0 , ad (fi ) 1−Aij ( f j)= 0 (47)A key property of every KM algebra is the triangular decompositiong(A) = n − ⊕ h ⊕ n + (48)where n + and n − , respectively, are spanned by the multiple commutatorsof the e i and f i which do not vanish on account of the Serre relations or the