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8.1 Lie groups and Lie algebras 93Equations (8.4) show that the group G has the same dimension r at allpoints, and that the set of all right-invariant vector fields and the tangentspace T q0 to G at q 0 are isomorphic vector spaces. An r-dimensional groupis denoted by G r and said to be of r parameters.The transformations Φ t generated by a right-invariant vector field v,in the way described in §2.8, clearly commute with right translations. IfΦ t q 0 = q(t), we findΦ t q ′ =Φ t R q ′q 0 = R q ′Φ t q 0 = R q ′q(t) =q(t)q ′ , (8.5)so that Φ t = L q(t) ; the right-invariant vector fields represent infinitesimalleft translations. From (2.25), the commutator of two right-invariantvector fields is also right-invariant, so that if we take a basis {ξ A ,A=1,...,r} of the space of right-invariant vector fields we must have[ξ A , ξ B ]=C C ABξ C , C C AB = −C C BA. (8.6)The coefficients C C AB are known as the structure constants of the group.A Lie algebra is defined to be a (finite-dimensional) vector space in whicha bilinear operation [u, v], obeying [u, v] =−[v, u] and the Jacobi identity(2.7), is defined. Thus we have provedTheorem 8.1 A Lie group defines a unique Lie algebra.It is possible to show that the converse also holds.Theorem 8.2 Every Lie algebra defines a unique (simply-connected) Liegroup.For a proof, see e.g. Cohn (1957). The elements of the Lie algebra, ora basis of them, are said to generate the group. Noting that the Jacobiidentity (2.7) holds for (8.6) if and only ifwe can rewrite Theorem 8.2 asC E [ABC F C]E =0, (8.7)Theorem 8.3 (Lie’s third fundamental theorem). Any set of constantsC A BC satisfying C A BC = C A [BC] and (8.7) are the structure constants ofa group.Theorem 8.2 does not imply that a given Lie algebra arises from onlyone Lie group. For example the Lorentz group L ↑ + and the group SL(2,C)(see §3.6) have the same Lie algebra. It is true, however, that all connectedLie groups with a given Lie algebra are homomorphic images of the onespecified in Theorem 8.2.