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.
Cartan's theorem
The equivalence between the category of simply connected real Lie groups and finite-dimensional real Lie algebras is called usually (in the literature of the second half of 20th century) Cartan's or Cartan-Lie theorem as it is proved by Élie Cartan whereas S. Lie has proved earlier just the infinitesimal version (local solvability of Maurer-Cartan equations (see Maurer-Cartan form) or the equivalence between the finite-dimensional Lie algebras and the category of local Lie groups). Lie listed his results as 3 direct and 3 converse theorems, the infinitesimal variant of Cartan's theorem was essentially his 3rd converse theorem, hence Serre has called it in an influential book, the "third Lie theorem", the name which is historically somewhat misleading, but more often used in the recent decade in the connection to many generalizations.
See also
References
- Cartan, Élie (1930), "La théorie des groupes finis et continus et l'Analysis Situs", Mémorial Sc. Math., XLII, pp. 1–61
- Helgason, Sigurdur (2001), Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, 34, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2848-9, MR 1834454