- Lie theory
Lie theory is an area of
mathematics , 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
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