In the mathematics of

Lie theory In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is ...

, Lie's third theorem states that every finite-dimensional Lie algebra

\mathfrak

over the real numbers is associated to a Lie group ''

G

''. The theorem is part of the

Lie group–Lie algebra correspondence In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are ...

. Historically, the third theorem referred to a different but related result. The two preceding theorems of

Sophus Lie Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. Life and career Marius Soph ...

, restated in modern language, relate to the

infinitesimal transformation In mathematics, an infinitesimal transformation is a limiting form of ''small'' transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 s ...

s of a

group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

on a

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

. The third theorem on the list stated the

Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the asso ...

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 In differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable. __TOC__ Definition and first properties Let \sigma: G \times M \to M, ( ...

. The result now known as the third theorem provides an intrinsic and global converse to the original theorem.

Historical notes

The equivalence between the

category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...

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 Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...

. 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 Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...

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 In abstract algebra, Ado's theorem is a theorem characterizing finite-dimensional Lie algebras. Statement Ado's theorem states that every finite-dimensional Lie algebra ''L'' over a field ''K'' of characteristic zero can be viewed as a Lie algebr ...

and another recounting the proof by Élie Cartan.

Proofs

There are several proofs of Lie's third theorem, each of them employing different algebraic and/or geometric techniques.

Algebraic proof

The classical proof is straighforward but relies on

, whose proof is algebraic and highly non-trivial. Ado's theorem states that any finite-dimensional Lie algebra can be represented by

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

. As a consequence, integrating such algebra of matrices via the

matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives ...

yields a Lie group integrating the original Lie algebra.

Cohomological proof

A more geometric proof is due to

and was published by . This proof uses

induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...

on the dimension of the

center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...

and it involves the Chevalley-Eilenberg complex.

Geometric proof

A different geometric proof was discovered in 2000 by Duistermaat and Kolk. Unlike the previous ones, it is a

constructive proof In mathematics, a constructive proof is a method of mathematical proof, proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also ...

: the integrating Lie group is built as the quotient of the (infinite-dimensional) Banach Lie group of paths on the Lie algebra by a suitable subgroup. This proof was influential for Lie theory since it paved the way to the generalisation of Lie third theorem for Lie groupoids and Lie algebroids.

References

* * *

External links

Encyclopaedia of Mathematics (EoM) article
{{Manifolds Lie algebras Lie groups Theorems about algebras