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(You can check your answers by comparing your results with Example15.3 on page 410 in the following chapter.)
15Homogeneous SpacesIn this section we will explore the concept of symmetries even further. We in-troduced some of the basics in chapter 6, and we will pursue the ideas furtherhere. In doing so, we will generalise the FRW models to the Bianchi modelswhich are in general homogeneous but not necessarily isotropic.15.1Lie groups and Lie algebrasFirst we will introduce some very important concepts used in mathematicsand physics. Whenever we talk about continuous symmetries, the words Liegroups and Lie algebra are usually mentioned.We saw earlier that the Killing vectors generate a special class of diffeo-morphisms; Killing vectors generateisometries. The isometries of a space forma group. For instance, let us take the sphere,S2, with the usual round met-ric. The isometry group of the sphere is all the rotations in three-dimensionalspace that leaves the sphere invariant. These (orientation-preserving1) rota-tions form the groupSO(3). What is so special about this group, is that thegroup itself, can be considered as a manifold!Since the dimension of thegroup is three, the groupSO(3)can be considered as a three-dimensionalmanifold. The groupSO(3)is an example of aLie group. We define Lie groupsas follows.Definition: Lie Group.A Lie group,G, is a topological space that has thefollowing properties:1.Gis a manifold.2. The group multiplicationm:G×Gmapsto−→Gis smooth.3. Inversioni:Gmapsto−→Gis smooth.1We will always assume that we are talking about orientation-preserving isometries, unlessstated otherwise.
402Homogeneous SpacesTo show thatSO(3)has these properties is not difficult. We already knowthat multiplication and inversion are continuous operations. Each element inSO(3)corresponds to a rotation, and rotations are continuous operations. Wecan show thatSO(3)is actually equal to the manifoldP3and hence,SO(3)isa manifold.Let us also define what we mean withLie algebra.Definition: Lie Algebra.A real (or complex) Lie algebra,g, is a (finite di-mensional) vector space equipped with a bilinear map[,] :g×gmapsto−→gwhich satisfies the following properties:1.[X,X] = 0for allXg.2. Jacobi’s identity:[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0(15.1)for allX,Y,Zg.Note that 1. implies that the bilinear map[,]is skew-symmetric:[X,Y] =[Y,X](15.2)An example of a Lie algebra is the space of alln×nmatricesgl(n). The bracket[,]is in this case simply defined by[A,B] =ABBA(15.3)for all matricesAandB.The bracket is in this case the usual commutatormultiplication of matrices.There is actually a deep connection between these two concepts.A Liealgebra is a vector space, while a Lie group is a group manifold. Amazinglywe have the following theorem.Theorem:LetGbe a Lie group.The tangent space ofGat the identityelement,TeG, is a Lie algebra. Hence,g=TeG.

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Term
Fall
Professor
JANECLON BANDAM
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