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Lie's third theorem
In mathematics, Lie's third theorem states that every finite-dimensional Lie algebra ''g'' over the real numbers is associated to a Lie group ''G''.
Historically, the third theorem referred to a different but related result. The two preceding theorems of Sophus Lie, restated in modern language, relate to the infinitesimal transformations of a transformation group acting on a smooth manifold. The third theorem on the list stated the Jacobi identity for the infinitesimal transformations of a local Lie group. Conversely, in the presence of a Lie algebra of vector fields, integration gives a ''local'' Lie group action. The result now known as the third theorem provides an intrinsic and global converse to the original theorem.
==External links==
*(Encyclopaedia of Mathematics (EoM) article at Springer.de )
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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