Lie theory
Encyclopedia
Lie theory is an area of mathematics
, developed initially by Sophus Lie
.
Early expressions of Lie theory are found in books composed by Lie with Friedrich Engel
and Georg Scheffers
from 1888 to 1896.
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 of modular form
s, in the hands of Felix Klein
and Henri Poincaré
. The initial application that Lie had in mind was to the theory of differential equation
s. On the model of Galois theory
and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry
, the whole area of ordinary differential equation
s.
The hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. 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 of quadrature
s, the indefinite integrals 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 in automorphic representations and in mathematical physics
, and the subject has become a busy crossroads.
s are the first instance of Lie theory. The compact
case arises through Euler's formula
in the complex plane
. Other one-parameter groups occur in the split-complex number
plane as the unit hyperbola
and in the dual number
plane as the line
In these cases the Lie algebra parameters have names: angle
, hyperbolic angle
, and slope
. Using the appropriate "angle", and a radial vector, any one of these planes can be given a polar decomposition. Any one of these decompositions, or Lie algebra renderings, may be necessary for rendering the Lie subalgebra of a 2 × 2 real matrix.
There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identified with the 3-sphere
. Its Lie algebra is the subspace of quaternion
vectors. Since the commutator
ij − ji = 2k, the Lie bracket in this algebra is twice the cross product
of ordinary vector analysis.
Another elementary 3-parameter example is given by the Heisenberg group and its Lie algebra.
Standard treatments of Lie theory often begin with the Classical group
s.
relating Lie algebra
to Lie group
. Structure is captured in Root system
s and root data
.
Lie theory is frequently built upon a study of the classical linear algebraic group
s. Special branches include Weyl group
s, Coxeter group
s, and buildings. The classical subject has been extended to Groups of Lie type
.
In 1900 David Hilbert
challenged Lie theorists with his Fifth Problem
presented at the International Congress of Mathematicians
in Paris.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, developed initially by Sophus Lie
Sophus Lie
Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...
.
Early expressions of Lie theory are found in books composed by Lie with Friedrich Engel
Friedrich Engel
Dr. Friedrich Wilhelm Konrad Siegfried Engel was a German SS officer who was convicted in absentia of 246 murder charges by an Italian military court in 1999 for his role in the 1944 execution of Italian captives in retalition for a partisan attack against German soldiers, and who as a result...
and Georg Scheffers
Georg Scheffers
Georg Scheffers was a German mathematician specializing in differential geometry. He was born on November 21, 1899 in the village of Altendorf near Holzminden . Scheffers began his university career at the University of Leipzig where he studied with Felix Klein and Sophus Lie...
from 1888 to 1896.
In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete group
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
s that had developed in the theory of modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s, in the hands of Felix Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
and Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
. The initial application that Lie had in mind was to the theory of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s. On the model of Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
, the whole area of ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s.
The hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory
Differential Galois theory
In mathematics, differential Galois theory studies the Galois groups of differential equations.Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of...
, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadrature
Quadrature
Quadrature may refer to:In signal processing:*Quadrature amplitude modulation , a modulation method of using both an carrier wave and a 'quadrature' carrier wave that is 90° out of phase with the main, or in-phase, carrier...
s, the indefinite integrals 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
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
. Some major areas of application have been found, for example in automorphic representations and in mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, and the subject has become a busy crossroads.
Elementary Lie theory
The one-parameter groupOne-parameter group
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphismfrom the real line R to some other topological group G...
s are the first instance of Lie theory. The compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
case arises through Euler's formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
. Other one-parameter groups occur in the split-complex number
Split-complex number
In abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the formwhere x and y are real numbers...
plane as the unit hyperbola
Unit hyperbola
In geometry, the unit hyperbola is the set of points in the Cartesian plane that satisfies x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial lengthWhereas the unit circle surrounds its center, the unit hyperbola requires the...
and in the dual number
Dual number
In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 . The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε with a and...
plane as the line
In these cases the Lie algebra parameters have names: angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
, hyperbolic angle
Hyperbolic angle
In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle...
, and slope
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....
. Using the appropriate "angle", and a radial vector, any one of these planes can be given a polar decomposition. Any one of these decompositions, or Lie algebra renderings, may be necessary for rendering the Lie subalgebra of a 2 × 2 real matrix.
There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identified with the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
. Its Lie algebra is the subspace of quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
vectors. Since the commutator
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
ij − ji = 2k, the Lie bracket in this algebra is twice the cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
of ordinary vector analysis.
Another elementary 3-parameter example is given by the Heisenberg group and its Lie algebra.
Standard treatments of Lie theory often begin with the Classical group
Classical group
In mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type...
s.
Aspects of Lie theory
The foundation of Lie theory is the exponential mapExponential map
In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....
relating Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
to Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
. Structure is captured in Root system
Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras...
s and root data
Root datum
In mathematics, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism...
.
Lie theory is frequently built upon a study of the classical linear algebraic group
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...
s. Special branches include Weyl group
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
s, Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
s, and buildings. The classical subject has been extended to Groups of Lie type
Group of Lie type
In mathematics, a group of Lie type G is a group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups...
.
In 1900 David Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
challenged Lie theorists with his Fifth Problem
Hilbert's fifth problem
Hilbert's fifth problem, is the fifth mathematical problem from the problem-list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups. The theory of Lie groups describes continuous symmetry in mathematics; its importance there and in theoretical physics...
presented at the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in Paris.