|
|
| Line 1: |
Line 1: |
| '''Lie theory''' ({{IPAc-en|ˈ|l|iː}} {{respell|LEE|'}}) is an area of [[mathematics]], developed initially by [[Sophus Lie]].
| | Hello. Allow me [http://www.Stdtestexpress.com/ introduce] the author. Her name is std home test home std test Emilia Shroyer but it's not the most feminine title std home test out there. Doing ceramics is what her family members and her appreciate. Hiring is her day job now and she will not alter it whenever quickly. For years he's been residing in North Dakota and his family members enjoys it.<br><br>Here is my webpage :: std testing at home ([http://www.muaguide.com/curing-candidiasis-get-done-easily/ click the following article]) |
| 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.
| |
| | |
| 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==
| |
| The [[one-parameter group]]s are the first instance of Lie theory. The [[compact space|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]]
| |
| :<math>\lbrace \exp(j t) = \cosh(t) + j \sinh(t) : t \in R \rbrace,</math>
| |
| and in the [[dual number]] plane as the line <math>\lbrace \exp(\epsilon t) = 1 + \epsilon t : t \in R \rbrace.</math>
| |
| 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#Alternative planar decompositions|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#Read a matrix|2 × 2 real matrix]].
| |
| | |
| There is a classical 3-parameter Lie group and algebra pair: the [[versor|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.
| |
| | |
| ==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]].
| |
| | |
| In 1900 [[David Hilbert]] challenged Lie theorists with his [[Hilbert's fifth problem|Fifth Problem]] presented at the [[International Congress of Mathematicians]] in Paris.
| |
| | |
| ==See also==
| |
| * [[List of Lie group topics]]
| |
| | |
| ==Further reading==
| |
| * [[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/}}
| |
| * [[J. L. Coolidge]] (1940) ''A History of Geometrical Methods'', pp 304–17, [[Oxford University Press]] ([[Dover Publications]] 2003).
| |
| * 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=David H. |last=Sattinger |first2=O. L. |last2=Weaver |year=1986 |title=Lie groups and algebras with applications to physics, geometry, and mechanics |publisher=Springer-Verlag |isbn=3-540-96240-9 }}
| |
| *{{cite book |authorlink=John Stillwell |first=John |last=Stillwell |year=2008 |title=Naive Lie Theory |publisher=Springer |isbn=0-387-98289-2 }}
| |
| * Heldermann Verlag [http://www.heldermann.de/JLT/jltcover.htm Journal of Lie Theory]
| |
| | |
| {{Mathematics-footer}}
| |
| | |
| [[Category:Lie groups]]
| |
| [[Category:Differential equations]]
| |
| [[Category:History of mathematics]]
| |
Hello. Allow me introduce the author. Her name is std home test home std test Emilia Shroyer but it's not the most feminine title std home test out there. Doing ceramics is what her family members and her appreciate. Hiring is her day job now and she will not alter it whenever quickly. For years he's been residing in North Dakota and his family members enjoys it.
Here is my webpage :: std testing at home (click the following article)