Lie group

A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction.

Lie's original motivation for introducing Lie groups was to model the continuous symmetries of differential equations, in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic equations.

Sophus Lie considered the winter of 1873–1874 as the birth date of his theory of continuous groups.

But during the 1870s all his papers (except the very first note) were published in Norwegian journals, which impeded recognition of the work throughout the rest of Europe.

[4] In 1884 a young German mathematician, Friedrich Engel, came to work with Lie on a systematic treatise to expose his theory of continuous groups.

Much of Jacobi's work was published posthumously in the 1860s, generating enormous interest in France and Germany.

Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries.

However, the hope that Lie theory would unify the entire field of ordinary differential equations was not fulfilled.

Additional impetus to consider continuous groups came from ideas of Bernhard Riemann, on the foundations of geometry, and their further development in the hands of Klein.

Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory: Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing, who in 1888 published the first paper in a series entitled Die Zusammensetzung der stetigen endlichen Transformationsgruppen (The composition of continuous finite transformation groups).

[7] The work of Killing, later refined and generalized by Élie Cartan, led to classification of semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using highest weights.

In 1900 David Hilbert challenged Lie theorists with his Fifth Problem presented at the International Congress of Mathematicians in Paris.

[8] The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley.

Lie groups play an enormous role in modern geometry, on several different levels.

Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant.

This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold.

On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of rigidity and yields a rich algebraic structure.

In the 1940s–1950s, Ellis Kolchin, Armand Borel, and Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field.

winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense subgroup of ⁠

Hilbert's fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples.

In two dimensions, if we restrict attention to simply connected groups, then they are classified by their Lie algebras.

We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows.

One important aspect of the study of Lie groups is their representations, that is, the way they can act (linearly) on vector spaces.

Assume the system in question has the rotation group SO(3) as a symmetry, meaning that the Hamiltonian operator

One can also study (in general infinite-dimensional) unitary representations of an arbitrary Lie group (not necessarily compact).

It turns out that they mostly fall into four infinite families, the "classical Lie algebras" An, Bn, Cn and Dn, which have simple descriptions in terms of symmetries of Euclidean space.

Write so that we have a sequence of normal subgroups Then This can be used to reduce some problems about Lie groups (such as finding their unitary representations) to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension.

The simplest way to define infinite-dimensional Lie groups is to model them locally on Banach spaces (as opposed to Euclidean space in the finite-dimensional case), and in this case much of the basic theory is similar to that of finite-dimensional Lie groups.

However this is inadequate for many applications, because many natural examples of infinite-dimensional Lie groups are not Banach manifolds.

Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces.