and contact of spheres that have come to be called Lie theory. 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...

uses structure theory of semisimple Lie algebras. For classical Lie algebras there is a more explicit construction. Let g0 be a real Lie algebra of matrices...

between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the representation theory of Lie groups...

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...

associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements...

Marius Sophus Lie (/liː/ LEE; Norwegian: [liː]; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous...

its Lie algebra; this correspondence is discussed in detail in subsequent sections. See representation of Lie algebras for the Lie algebra theory. In...

In the theory of Lie groups, the exponential map is a map from the Lie algebra g {\displaystyle {\mathfrak {g}}} of a Lie group G {\displaystyle G} to...

formula in Lie group theory. As the 3-sphere represented by versors in   H   {\displaystyle \ \mathbb {H} \ } is a 3-parameter Lie group, practice...

contributions to the fields of probability and algebra, especially semisimple Lie groups, Lie algebras, and Markov processes. The Dynkin diagram, the Dynkin system...

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...

applications in probability theory. Ergodic theory has fruitful connections with harmonic analysis, Lie theory (representation theory, lattices in algebraic...

metric, and implies that the Lie algebra is the Lie algebra of a compact Lie group; conversely, any compact (or abelian) Lie group has such a Riemannian...

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...

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...

Local Lie group Formal group law Hilbert's fifth problem Hilbert-Smith conjecture Lie group decompositions Real form (Lie theory) Complex Lie group Complexification...

infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras, and differential geometry are used to understand...

representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out...

Intersection theory — Invariant theory — Iwasawa theory — K-theory — KK-theory — Knot theory — L-theory — Lie theory — Littlewood–Paley theory — Matrix theory —...

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...

Hofmann; Sidney A. Morris (2007). The Lie Theory of Connected Pro-Lie Groups: A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups, and Connected Locally Compact...

infinitesimal generators do also; the Lie bracket [Di, Dj] = 0 is this property's reflection. In other words, the Lie derivative of one coordinate with respect...

In terms of Lie theory, the Rodrigues' formula provides an algorithm to compute the exponential map from the Lie algebra so(3) to its Lie group SO(3)...

fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and...

the action of T. In Lie theory, a one-dimensional subspace of a Lie algebra is mapped to a one-parameter group under the Lie group–Lie algebra correspondence...

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...

K-theory Knot theory L-theory Lattice theory Lie theory M-theory Measure theory Model theory Morse theory Module theory Nevanlinna theory Number theory...