Ivancevic_Applied-Diff-Geom

264 Applied Differential Geometry: A Modern Introduction3.9.5.2 Maurer–Cartan 1–FormsLet be given a Lie group G with the Lie algebra G ≃ T e (G), whose basisis a set {A i ∈ G : i = 1, ..., r}, where r = dim G ≡ dim G. Let also aset U 0 ⊂ {a i ∈ R : i = 1, ..., r} be some open neighborhood of the zeropoint in R r . The exponential mapping exp : U 0 → G 0 , where by definition[Blackmore et. al. (1998)]R r ⊃ U 0 ∋ (a 1 , . . . , a r ) :exp ✲ exp(a i A i)= a ∈ G0 ⊂ G, (3.85)is an analytical mapping of the whole U 0 on some open neighborhood G 0of the unity element e ∈ G. From (3.85) it is easy to find that T e (G) =T e (G 0 ) ≃ G, where e = exp(0) ∈ G. Define now the following left–invariantG−valued differential 1–form on G 0 ⊂ G:¯ω(a, da) = a −1 da = ¯ω j (a, da)A j , (3.86)where A j ∈ G, ¯ω j (a, da) ∈ T ∗ a (G), a ∈ G 0 , j = 1, ..., r. To build effectivelythe unknown forms {¯ω j (a, da) : j = 1, ..., r}, let us consider the followinganalytical one–parameter 1–form ¯ω t (a, da) = ¯ω(a t ; da t ) on G 0 , wherea t = exp ( ta i A i), t ∈ [0, 1], and differentiate this form with respect to theparameter t ∈ [0, 1]. We will get [Blackmore et. al. (1998)]d¯ω t /dt = −a j A j a −1tda t +a −1t a t da j A j +a −1t da t a j A j = −a j [A j , ¯ω t ]+A j da j .(3.87)Having used the Lie identity [A j , A k ] = c i jk A i, j, k = 1, ..., r, and the righthand side of (3.86) in formwe ultimately get that¯ω j (a, da) = ¯ω j k (a)dak , (3.88)ddt (t¯ωj i (ta)) = Aj k t¯ωk i (ta) + δ j i , (3.89)where the matrix A k i , i, k = 1, ..., r, is defined as follows:A k i = c k ija j . (3.90)Thus, the matrix W j i (t) = t¯ωj i (ta), i, j = 1, ..., r, satisfies the following from(3.89) differential equation [Chevalley (1955)]dW/dt = AW + E, W | t=0= 0, (3.91)