Abstract
(3) Thus a,(O) = 0, and at the origin the Jacobian matrix of a, can be found from (3), ai. = A*-‘(O) A&,(O). 0 ne may take A,(O) = Z and ai* = Ai. Let L = {A,,***,Ak}LA, the real Lie algebra generated by the matrices A i, and 9 = {a, ,..., ukILA, the Lie algebra of vector fields on R” generated by the a,. Correspondence (3) preserves the Lie bracket, so that A i -+ a, gives an algebra isomorphism of the generated Lie algebras. In this case 9 is said to be faithful to its linear Lie algebra
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