Difference between revisions of "Lie theory"

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'''Lie theory''' ({{IPAc-en|ˈ|l|iː}} {{respell|LEE|'}}) is an area of [[mathematics]], developed initially by [[Sophus Lie]].
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'''Lie theory''' ({{IPAc-en|ˈ|l|iː}} {{respell|LEE|'}}) is one of the [[areas of mathematics]], developed initially by [[Sophus Lie]] and worked out by [[Wilhelm Killing]] and [[Élie Cartan]]. The foundation of Lie theory is the [[exponential map (Lie theory)|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 manifold]]s. Lie groups evolve out of the identity (1) and the [[tangent vector]]s to [[one-parameter group|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 system]]s and [[Root datum|root data]].
Early expressions of Lie theory are found in books composed by Lie with [[Friedrich Engel (mathematician)|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 equation]]s, the driving conception was of a theory capable of unifying, by the study of [[symmetry]], the whole area of [[ordinary differential equation]]s.
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Lie theory has been particularly useful in [[mathematical physics]] since it describes important physical groups such as the [[Galilean transformation#Galilean group|Galilean group]], the [[Lorentz group]] and the [[Poincaré group]].
 
 
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 (mathematics)|quadrature]]s, the [[indefinite integral]]s 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 representation]]s and in [[mathematical physics]], and the subject has become a busy crossroads.
 
  
 
==Elementary Lie theory==
 
==Elementary Lie theory==
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Another elementary 3-parameter example is given by the [[Heisenberg group]] and its Lie algebra.
 
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.
 
Standard treatments of Lie theory often begin with the [[Classical group]]s.
 +
 +
==History and scope==
 +
Early expressions of Lie theory are found in books composed by [[Sophus Lie]] with [[Friedrich Engel (mathematician)|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 equation]]s, the driving conception was of a theory capable of unifying, by the study of [[symmetry]], the whole area of [[ordinary differential equation]]s.
 +
 +
According to historian Thomas W. Hawkins, it was [[Elie Cartan]] that made Lie theory what it is:
 +
:While Lie had many fertile ideas, Cartan was primarily responsible for the extensions and applications of his theory that have made it a basic component of modern mathematics. It was he who, with some help from [[Hermann Weyl|Weyl]], developed the seminal, essentially algebraic ideas of [[Wilhelm Killing|Killing]] into the theory of the structure and representation of [[semisimple Lie algebra]]s that plays such a fundamental role in present-day Lie theory. And although Lie envisioned applications of his theory to geometry, it was Cartan who actually reated them, for example through his theories of symmetric and generalized spaces, including all the attendant apparatus ([[moving frame]]s, [[exterior differential form]]s, etc.)<ref>Thomas Hawkins (1996) [[Historia Mathematica]] 23(1):92–5</ref>
  
 
==Aspects of Lie theory==
 
==Aspects of Lie theory==
The foundation of Lie theory is the [[exponential map]] relating [[Lie algebra]] to [[Lie group]]. Structure is captured in [[Root system]]s and [[Root datum|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 [[Bruhat-Tits building|buildings]]. The classical subject has been extended to [[Group of Lie type|Groups of Lie type]].
 
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 [[Bruhat-Tits building|buildings]]. The classical subject has been extended to [[Group of Lie type|Groups of Lie type]].
  
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==See also==
 
==See also==
 
* [[List of Lie group topics]]
 
* [[List of Lie group topics]]
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==Notes and references==
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{{Reflist}}
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* John A. Coleman (1989) "The Greatest Mathematical Paper of All Time," [[The Mathematical Intelligencer]] 11(3):&nbsp;29–38.
  
==Further reading==
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== Further reading ==
 +
* M.A. Akivis & B.A. Rosenfeld (1993) ''Élie Cartan (1869&ndash;1951)'', translated from Russian original by V.V. Goldberg, chapter 2: Lie groups and Lie algebras, [[American Mathematical Society]] ISBN 0-8218-4587-X .
 
* [[P. M. Cohn]] (1957) ''Lie Groups'', Cambridge Tracts in Mathematical Physics.
 
* [[P. M. Cohn]] (1957) ''Lie Groups'', Cambridge Tracts in Mathematical Physics.
** {{cite journal|author=Nijenhuis, Albert|authorlink=Albert Nijenhuis|title=Review: ''Lie groups'', by P. M. Cohn|journal=[[Bulletin of the American Mathematical Society]]|year=1959|volume=1959|volume=65|issue=6|pages=338–341|url=http://www.ams.org/journals/bull/1959-65-06/S0002-9904-1959-10358-X/}}
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** {{cite journal |author=Nijenhuis, Albert |authorlink=Albert Nijenhuis |title=Review: ''Lie groups'', by P. M. Cohn |journal=[[Bulletin of the American Mathematical Society]] |year=1959 |volume=65 |issue=6 |pages=338–341 |url=http://www.ams.org/journals/bull/1959-65-06/S0002-9904-1959-10358-X/ |doi=10.1090/s0002-9904-1959-10358-x}}
 
* [[J. L. Coolidge]] (1940) ''A History of Geometrical Methods'', pp 304–17, [[Oxford University Press]] ([[Dover Publications]] 2003).
 
* [[J. L. Coolidge]] (1940) ''A History of Geometrical Methods'', pp 304–17, [[Oxford University Press]] ([[Dover Publications]] 2003).
 +
* Robert Gilmore (2008) ''Lie groups, physics, and geometry: an introduction for physicists, engineers and chemists'', [[Cambridge University Press]] ISBN 9780521884006 .
 
* F. Reese Harvey (1990) ''Spinors and calibrations'', [[Academic Press]], ISBN 0-12-329650-1 .
 
* F. Reese Harvey (1990) ''Spinors and calibrations'', [[Academic Press]], ISBN 0-12-329650-1 .
 
*{{cite book |first=Thomas |last=Hawkins |year=2000 |title=Emergence of the Theory of Lie Groups: an essay in the history of mathematics, 1869–1926 |publisher=Springer |isbn=0-387-98963-3 }}
 
*{{cite book |first=Thomas |last=Hawkins |year=2000 |title=Emergence of the Theory of Lie Groups: an essay in the history of mathematics, 1869–1926 |publisher=Springer |isbn=0-387-98963-3 }}
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* Heldermann Verlag [http://www.heldermann.de/JLT/jltcover.htm Journal of Lie Theory]
 
* Heldermann Verlag [http://www.heldermann.de/JLT/jltcover.htm Journal of Lie Theory]
  
{{Mathematics-footer}}
 
  
[[Category:Lie groups]]
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{{Areas of mathematics}}
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[[Category:Lie groups| ]]
 
[[Category:Differential equations]]
 
[[Category:Differential equations]]
 
[[Category:History of mathematics]]
 
[[Category:History of mathematics]]

Latest revision as of 01:28, 2 January 2015

Lie theory (Template:IPAc-en Template:Respell) is one of the areas of mathematics, developed initially by Sophus Lie and 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 important physical groups such as the Galilean group, the Lorentz group and the Poincaré group.

Elementary Lie theory

The one-parameter groups 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 groups.

History and scope

Early expressions of Lie theory are found in books composed by Sophus 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 groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. 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 equations.

According to historian Thomas W. Hawkins, it was Elie Cartan that made Lie theory what it is:

While Lie had many fertile ideas, Cartan was primarily responsible for the extensions and applications of his theory that have made it a basic component of modern mathematics. It was he who, with some help from Weyl, developed the seminal, essentially algebraic ideas of Killing into the theory of the structure and representation of semisimple Lie algebras that plays such a fundamental role in present-day Lie theory. And although Lie envisioned applications of his theory to geometry, it was Cartan who actually reated them, for example through his theories of symmetric and generalized spaces, including all the attendant apparatus (moving frames, exterior differential forms, etc.)[1]

Aspects of Lie theory

Lie theory is frequently built upon a study of the classical linear algebraic groups. Special branches include Weyl groups, Coxeter groups, 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.

See also

Notes and references

  1. Thomas Hawkins (1996) Historia Mathematica 23(1):92–5

Further reading

  • M.A. Akivis & B.A. Rosenfeld (1993) Élie Cartan (1869–1951), translated from Russian original by V.V. Goldberg, chapter 2: Lie groups and Lie algebras, American Mathematical Society ISBN 0-8218-4587-X .
  • P. M. Cohn (1957) Lie Groups, Cambridge Tracts in Mathematical Physics.
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