- Lie theory
Lie theory is an area of
mathematics, developed initially by Sophus Lie.
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 Kleinand Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theoryand polynomial equations, the driving conception was of a theory capable of unifying by the study of symmetrythe whole area of ordinary differential equations.
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 of quadratures, the indefinite integrals 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 in automorphic representations and in mathematical physics, and the subject has become a busy crossroads.
Aspects of Lie theory
Here is a short list of objects of study:
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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