Lie's third theorem
Encyclopedia
In mathematics
, Lie's third theorem often means the result that states that any finite-dimensional Lie algebra
g, over the real numbers, is the Lie algebra associated to some Lie group
G. The relationship to the history has though become confused.
There were (naturally) two other preceding theorems, of Sophus Lie
. Those relate to the infinitesimal transformation
s of a transformation group acting on a smooth manifold. But, in fact, that language is anachronistic. The manifold concept was not clearly defined at the time, the end of the nineteenth century, when Lie was founding the theory. The conventional third theorem on the list was a result stating the Jacobi identity
for the infinitesimal transformations, of a local Lie group. This result has a converse, stating that in the presence of a Lie algebra of vector fields, integration gives a local Lie group action. The result initially stated is an intrinsic and global converse to the original theorem, therefore.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Lie's third theorem often means the result that states that any finite-dimensional Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
g, over the real numbers, is the Lie algebra associated to some Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
G. The relationship to the history has though become confused.
There were (naturally) two other preceding theorems, of Sophus Lie
Sophus Lie
Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
. Those relate to the infinitesimal transformation
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 skew-symmetric matrix A...
s of a transformation group acting on a smooth manifold. But, in fact, that language is anachronistic. The manifold concept was not clearly defined at the time, the end of the nineteenth century, when Lie was founding the theory. The conventional third theorem on the list was a result stating the Jacobi identity
Jacobi identity
In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...
for the infinitesimal transformations, of a local Lie group. This result has a converse, stating that in the presence of a Lie algebra of vector fields, integration gives a local Lie group action. The result initially stated is an intrinsic and global converse to the original theorem, therefore.