# Lie theory

In mathematics, the investigator Sophus Lie (/ˈl/ LEE) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory.[1] For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan.

The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data.

Lie theory has been particularly useful in mathematical physics since it describes important physical groups such as the Galilean group, the Lorentz group and the Poincaré group.

## Elementary Lie theory

The one-parameter groups are the first instance of Lie theory. The compact case arises through Euler's formula in the complex plane. Other one-parameter groups occur in the split-complex number plane as the unit hyperbola

$\lbrace \exp(j t) = \cosh(t) + j \sinh(t) : t \in R \rbrace,$

and in the dual number plane as the line $\lbrace \exp(\epsilon t) = 1 + \epsilon t : t \in R \rbrace.$ In these cases the Lie algebra parameters have names: angle, hyperbolic angle, and slope. Using the appropriate "angle", and a radial vector, any one of these planes can be given a polar decomposition. Any one of these decompositions, or Lie algebra renderings, may be necessary for rendering the Lie subalgebra of a 2 × 2 real matrix.

There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identified with the 3-sphere. Its Lie algebra is the subspace of quaternion vectors. Since the commutator ij − ji = 2k, the Lie bracket in this algebra is twice the cross product of ordinary vector analysis.

Another elementary 3-parameter example is given by the Heisenberg group and its Lie algebra. Standard treatments of Lie theory often begin with the Classical groups.

## History and scope

Early expressions of Lie theory are found in books composed by Sophus Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896.

In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations.

According to historian Thomas W. Hawkins, it was Élie Cartan that made Lie theory what it is:

While Lie had many fertile ideas, Cartan was primarily responsible for the extensions and applications of his theory that have made it a basic component of modern mathematics. It was he who, with some help from Weyl, developed the seminal, essentially algebraic ideas of Killing into the theory of the structure and representation of semisimple Lie algebras that plays such a fundamental role in present-day Lie theory. And although Lie envisioned applications of his theory to geometry, it was Cartan who actually created them, for example through his theories of symmetric and generalized spaces, including all the attendant apparatus (moving frames, exterior differential forms, etc.)[2]

## Aspects of Lie theory

Lie theory is frequently built upon a study of the classical linear algebraic groups. Special branches include Weyl groups, Coxeter groups, and buildings. The classical subject has been extended to Groups of Lie type.

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

## Notes and references

1. "Lie’s lasting achievements are the great theories he brought into existence. However, these theories – transformation groups, integration of differential equations, the geometry of contact – did not arise in a vacuum. They were preceded by particular results of a more limited scope, which pointed the way to more general theories that followed. The line-sphere correspondence is surely an example of this phenomenon: It so clearly sets the stage for Lie’s subsequent work on contact transformations and symmetry groups." R. Milson (2000) "An Overview of Lie’s line-sphere correspondence", pp 1–10 of The Geometric Study of Differential Equations, J.A. Leslie & T.P. Robart editors, American Mathematical Society ISBN 0-8218-2964-5 , quotation pp 8,9
2. Thomas Hawkins (1996) Historia Mathematica 23(1):92–5