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 Élie 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 Élie 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)
Includes
Source Wikipedia: Andrew M. Gleason
Source Élie Cartan (Wikipedia)
Source Ernst Witt, Biography (MacTutor History of Mathematics)
Source Kostant's convexity theorem - Wikipedia
Source Lie's theorem, Wikipedia
Source Maurer-Cartan Form (Wikipedia)
Source Wikipedia: Orthogonal group
Source Poincare-Birkhoff-Witt theorem (Wikipedia)
Source Sophus Lie (Wikipedia)
Source Weyl group (Wikipedia)
Source Wilhelm Killing (Wikipedia)
Source Wilhelm Magnus, Biography (MacTutor History of Mathematics)
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 passage
Élie Cartan's 1894 dissertation was essentially a rigorous rewriting of Killing's paper.
Includes: Wilhelm Killing, Lead paragraph
Wilhelm Karl Joseph Killing (10 May 1847-11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.
View the Source Maurer-Cartan Form (Wikipedia)
Poincare-Birkhoff-Witt theorem (Wikipedia)
Élie Cartan (Wikipedia)
Includes: Élie Cartan, Lead paragraph [in-branch]Quote, Includes: Élie Cartan, Lead paragraph [in-branch]
the theory of Lie groups
Sophus Lie (Wikipedia)
Includes: Sophus Lie, Lead paragraph [in-branch]Quote, Includes: Sophus Lie, Lead paragraph [in-branch]
the theory of continuous symmetry
Wikipedia: Andrew M. Gleason
Includes: Andrew M. Gleason, LeadQuote, Includes: Andrew M. Gleason, Lead
Andrew Mattei Gleason (November 4, 1921, October 17, 2008) was an American mathematician who made fundamental contributions to widely varied areas of mathematics, including the solution of Hilbert's fifth problem, and was a leader in reform and innovation in mathematics teaching at all levels.
Weyl group (Wikipedia)
Includes: Weyl Group, Lead paragraphQuote, Includes: Weyl Group, Lead paragraph
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system.
Kostant's convexity theorem - Wikipedia
Includes: Kostant's Convexity Theorem, Lead paragraphQuote, Includes: Kostant's Convexity Theorem, Lead paragraph
Konstant's convexity theorem states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is a convex set.
Wikipedia: Orthogonal group
Includes: Orthogonal Group, Lead paragraphQuote, Includes: Orthogonal Group, Lead paragraph
The orthogonal group is an algebraic group and a Lie group.
Lie's theorem, Wikipedia
Includes: Lie's Theorem, Lead sectionQuote, Includes: Lie's Theorem, Lead section
In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically
Ernst Witt, Biography (MacTutor History of Mathematics)
Includes: Ernst Witt, SummaryQuote, Includes: Ernst Witt, Summary
Ernst Witt was a German mathematician and a leading algebraist. He proved important results on Lie algebras.
Wilhelm Magnus, Biography (MacTutor History of Mathematics)
Includes: Wilhelm Magnus, SummaryQuote, Includes: Wilhelm Magnus, Summary
Wilhelm Magnus was a German American mathematician who worked in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations.
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