Lie theory is the body of mathematics initiated by Sophus Lie, who began lines of study involving the integration of differential equations, automorphism groups and the contact of spheres that have come to be called Lie theory. The field centers on transformation groups and was further developed by Wilhelm Killing and Elie Cartan, with Lie sphere geometry, the study of the contact relationships between spheres, among its applications.
Facts
Central QuestionLie theory asks how continuous symmetry can be studied algebraically, through the correspondence between a Lie group and the Lie algebra formed by its tangent vectors at the identity. 1 Key DebateCredit for classifying the complex finite-dimensional simple Lie algebras is split between two mathematicians: Wilhelm Killing carried out the classification from 1888 to 1890, introducing the Cartan subalgebra and root systems along the way, but it was Elie Cartan's 1894 dissertation, a rigorous rewriting of Killing's paper, that gave the classification its first fully rigorous treatment. 2 Classification
Pure or Applied Connections
Associated With
Source Lie Theory (Wikipedia)
Sources
1. Lie Theory (Wikipedia)
Wikimedia FoundationLead section
The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group-Lie algebra correspondence.
Associated With: Geometry, Wikipedia article body, Lie theory, first section
The subject is part of differential geometry since Lie groups are differentiable manifolds.
View the Source 2. Wilhelm Killing (Wikipedia)
Wikimedia FoundationLead section, classification passageQuote, Lead section, classification passage
Élie Cartan's 1894 dissertation was essentially a rigorous rewriting of Killing's paper.
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