- Lie theory
**Lie theory**is an area ofmathematics , developed initially bySophus Lie .In Lie's early work, the idea was to construct a theory of "continuous groups", to complement the theory of

discrete group s that had developed in the theory ofmodular form s, in the hands ofFelix Klein andHenri Poincaré . The initial application that Lie had in mind was to the theory ofdifferential equation s. On the model ofGalois theory andpolynomial equation s, the driving conception was of a theory capable of unifying by the study ofsymmetry the whole area ofordinary differential equation s.This hope was not fulfilled, at least in the terms apparently hoped for. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a

differential Galois theory , but it was developed by others, such as Picard and Vessiot, and it provides a theory ofquadrature s, theindefinite integral s required to express solutions.In the longer term, it has not been the direct application of continuous symmetry to geometric questions that has made Lie theory a central chapter of contemporary mathematics. The fact that there is a good structure theory for Lie groups and their representations has made them integral to large parts of

abstract algebra . Some major areas of application have been found, for example inautomorphic representation s and inmathematical physics , and the subject has become a busy crossroads.**Aspects of Lie theory**Here is a short list of objects of study:

*

Lie algebra s

*Lie group s

*Linear algebraic group s

* Groups of Lie type

*Universal enveloping algebra s

*Coxeter group s/Weyl group s

*Root system s and Root data

* Buildings

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Lie theory**— Lie theory, = Lie algebra. (Cf. ↑Lie algebra) … Useful english dictionary**Journal of Generalized Lie Theory and Applications**— Infobox Journal title = Journal of Generalized Lie Theory and Applications discipline = mathematics journal abbreviation = publisher = country = Estonia history = founded 2008 website = http://www.jglta.astralgo.eu/ ISSN = 1736 4337 | 1736… … Wikipedia**Real form (Lie theory)**— Lie groups … Wikipedia**Lie-Theorie**— Die Lie Theorie ist in der Mathematik eine Theorie, die sich mit dem Lösen von Differentialgleichungen beschäftigt. Sie wurde von Sophus Lie in den 1870er und den 1880er Jahren begründet. Die Lie Gruppen und die Lie Algebra haben sich aus der Lie … Deutsch Wikipedia**Lie group**— Lie groups … Wikipedia**Lie algebra representation**— Lie groups … Wikipedia**Lie sphere geometry**— is a geometrical theory of planar or spatial geometry in which the fundamental concept is the circle or sphere. It was introduced by Sophus Lie in the nineteenth century. [The definitive modern textbook on Lie sphere geometry is Harvnb|Cecil|1992 … Wikipedia**Lie Yukou**— (zh cpw|c=列圄寇/列禦寇|p=Lìe Yǔkòu| w= Lieh Yü k ou ; fl. ca. 400 BCE) is considered the author of the Daoist book Liezi , which uses his honorific name Liezi (列子; Lièzĭ; Lieh tzu; Master Lie ). The second Chinese character in Yukou is written kou 寇… … Wikipedia**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… … Wikipedia**Lie, Sophus**— ▪ Norwegian mathematician born Dec. 17, 1842, Nordfjordeid, Norway died Feb. 18, 1899, Kristiania Norwegian mathematician who founded the theory of continuous groups and their applications to the theory of differential equations (differential… … Universalium