New Paths Towards Quantum Gravity (Lecture Notes in Physics, 807)

248 I.G. AvramidiFor the lack of a better name we call the algebra G the curvature algebra. Asitwill be clear from the next section it is a subalgebra of the total isometry algebra ofthe symmetric space. It should be clear that the holonomy algebra H is the subalgebraof the curvature algebra G. The curvature algebra G is compact; it is a direct sumof two ideals, G = G 0 ⊕G s , an Abelian center G 0 of dimension n 0 and a semi-simplealgebra G s of dimension p + n s .Next, we define a symmetric nondegenerate N × N matrix(γ AB ) =( )δab 0. (283)0 β ikThis matrix and its inverse γ AB will be used to lower and to raise the capital Latinindices.4.7.3.2 Killing Vectors FieldsWe will use extensively the isometries of the symmetric space M. We follow theapproach developed in [7, 3, 10, 13]. The generators of isometries are the Killingvector fields ξ. The set of all Killing vector fields forms a representation of theisometry algebra, the Lie algebra of the isometry group of the manifold M. Wedefine two subspaces of the isometry algebra. One subspace is formed by Killingvectors (called translations) satisfying the initial conditions ∇ μ ξ ν∣ ∣x=x ′ = 0 andanother subspace is formed by Killing vectors (called rotations) satisfying the initialconditions ξ ν∣ ∣x=x ′ = 0 .One can easily show that a basis of translations can be chosen as(√ √ ) b ∂P a = K cot K a∂y b , (284)where K = (K a b) is a matrix defined byK a b = R a cbd y c y d . (285)We can also show that the vector fieldsL i =−D b iay a ∂∂yb, (286)define p linearly independent rotations. By adding the trivial Killing vectors for flatsubspaces we find that the number of independent rotations is p + n 0 n s + n 0 (n 0 −1)/2 . We introduce the following notation (ξ A ) = (P a , L i ).By using the explicit form of the Killing vector fields obtained above [7] one canprove the following theorem.Theorem 2 The Killing vector fields ξ A form a representation of the curvaturealgebra G