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

60 H. Nicolaifor indefinite A. It is sometimes convenient to introduce the operation oftransposition acting on any Lie algebra element E asE T := −θ(E) (51)The subalgebra k is thus generated by the ‘anti-symmetric’ elements satisfyingE T = −E; after exponentiation, the elements of the maximally compactsubgroup K formally appear as ‘orthogonal matrices’ obeying k T = k −1 .Often one uses a so-called Cartan-Weyl basis for g(A). Using Greekindices µ, ν, . . . to label the root components corresponding to an arbitrarybasis H µ in the CSA, with the usual summation convention and aLorentzian metric G µν for an indefinite g, we have h i := α µ i H µ, where α µ iare the ‘contravariant components’, G µν α ν i ≡ α i µ , of the simple roots α i(i = 1, . . . r), which are linear forms on the CSA, with ‘covariant components’defined as α i µ ≡ α i (H µ ). To an arbitrary root α there corresponds aset of Lie-algebra generators E α,s , where s = 1, . . . , mult (α) labels the (ingeneral) multiple Lie-algebra elements associated with α. The root multiplicitymult (α) is always one for finite dimensional Lie algebras, and alsofor the real (= positive norm) roots, but generically grows exponentialllywith −α 2 for indefinite A. In this notation, the remaining Chevalley-Serregenerators are given by e i := E αi and f i := E −αi . Then,and[H µ , E α,s ] = α µ E α,s (52)[E α,s , E α′ ,t] = ∑ uc s,t,uαα ′ E α+α ′ ,u (53)The elements of the Cartan-Weyl basis are normalized such that〈H µ |H ν 〉 = G µν , 〈E α,s |E β,t 〉 = δ st δ α+β,0 (54)where we have assumed that the basis satisfies E T α,s = E −α,s . Let us finallyrecall that the Weyl group of a KM algebra is the discrete group generatedby reflections in the hyperplanes orthogonal to the simple roots.5. The Hyperbolic Kac Moody Algebra AE 3As we explained, the known symmetries of Einstein’s theory for specialtypes of solutions include the Ehlers and Matzner Misner SL(2, R) symmetries,which can be combined into the Geroch group SL(2, ̂ R)ce . Furthermore,in the reduction to one time dimension, Einstein’s theory is invariantunder a rigid SL(3, R) symmetry acting on the spatial dreibein. Hence, any