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 allX∈g.2. Jacobi’s identity:[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0(15.1)for allX,Y,Z∈g.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] =AB−BA(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.