Study of Lie groups, Lie algebras and differential equations
In mathematics, the mathematician Sophus Lie (/liː/LEE) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory.[1] For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan.
The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by root systems and root data.
Lie theory has been particularly useful in mathematical physics since it describes the standard transformation groups: the Galilean group, the Lorentz group, the Poincaré group and the conformal group of spacetime.
^"Lie’s lasting achievements are the great theories he brought into existence. However, these theories – transformation groups, integration of differential equations, the geometry of contact – did not arise in a vacuum. They were preceded by particular results of a more limited scope, which pointed the way to more general theories that followed. The line-sphere correspondence is surely an example of this phenomenon: It so clearly sets the stage for Lie’s subsequent work on contact transformations and symmetry groups." R. Milson (2000) "An Overview of Lie’s line-sphere correspondence", pp 1–10 of The Geometric Study of Differential Equations, J.A. Leslie & T.P. Robart editors, American Mathematical Society ISBN 0-8218-2964-5 , quotation pp 8,9
and contact of spheres that have come to be called Lietheory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach...
beyond these origins. Lie groups are named after Norwegian mathematician Sophus Lie (1842–1899), who laid the foundations of the theory of continuous transformation...
methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced...
In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational...
Marius Sophus Lie (/liː/ LEE; Norwegian: [liː]; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous...
mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices...
calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent...
its Lie algebra; this correspondence is discussed in detail in subsequent sections. See representation of Lie algebras for the Lie algebra theory. In...
a complex Lie group. Much of the theory of algebraic groups was developed by analogy with Lie groups. There are several reasons why a Lie group may not...
contributions to the fields of probability and algebra, especially semisimple Lie groups, Lie algebras, and Markov processes. The Dynkin diagram, the Dynkin system...
may be common in the theory of Lie algebras (qua vector spaces) and algebraic groups (qua algebraic varieties). In Galois theory, given a field extension...
Campbell–Baker–Hausdorff formula in Lie group theory. As the 3-sphere represented by versors in H {\displaystyle \mathbb {H} } is a 3-parameter Lie group, practice with...
knots is essentially unique. A maximal compact subgroup of a semisimple Lie group may not be unique, but is unique up to conjugation. An object that...
applications in probability theory. Ergodic theory has fruitful connections with harmonic analysis, Lietheory (representation theory, lattices in algebraic...
Intersection theory — Invariant theory — Iwasawa theory — K-theory — KK-theory — Knot theory — L-theory — Lietheory — Littlewood–Paley theory — Matrix theory —...
infinitesimal transformations of solutions to solutions (Lietheory). Continuous group theory, Lie algebras, and differential geometry are used to understand...
Hofmann; Sidney A. Morris (2007). The LieTheory of Connected Pro-Lie Groups: A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact...
transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group...
Manual of Mental Disorders (DSM). Various theories have been proposed to explain the causes of pathological lying, including stress, an attempt to shift...
topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include the circle group T...
In mathematics, Theory of Lie groups is a series of books on Lie groups by Claude Chevalley (1946, 1951, 1955). The first in the series was one of the...
retirement in 1993. Kostant's work has involved representation theory, Lie groups, Lie algebras, homogeneous spaces, differential geometry and mathematical...
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