Lie's third theorem

In the mathematics of Lie theory, Lie's third theorem states that every finite-dimensional Lie algebra over the real numbers is associated to a Lie group . The theorem is part of the Lie group–Lie algebra correspondence. 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 group action 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. The equivalence between the of simply connected real Lie groups and finite-dimensional real Lie algebras is usually called (in the literature of the second half of 20th century) Cartan's or the Cartan-Lie theorem as it was proved by Élie Cartan. Sophus Lie had previously proved the infinitesimal version: local solvability of the Maurer-Cartan equation, or the equivalence between the category of finite-dimensional Lie algebras and the category of local Lie groups. Lie listed his results as three direct and three converse theorems. The infinitesimal variant of Cartan's theorem was essentially Lie's third converse theorem. In an influential book Jean-Pierre Serre called it the third theorem of Lie. The name is historically somewhat misleading, but often used in connection to generalizations. Serre provided two proofs in his book: one based on Ado's theorem and another recounting the proof by Élie Cartan. There are several proofs of Lie's third theorem, each of them employing different algebraic and/or geometric techniques. The classical proof is straightforward but relies on Ado's theorem, whose proof is algebraic and highly non-trivial. Ado's theorem states that any finite-dimensional Lie algebra can be represented by matrices.