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1. Lie Algebra - Wikipedia, The Free Encyclopedia
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.
http://en.wikipedia.org/wiki/Lie_algebra
##### Lie algebra
Classical groups
... General linear group GL(n)
Special linear group
SL(n)
Orthogonal group
O(n)
Special orthogonal group
SO(n)
Unitary group
U(n)
Special unitary group
SU(n)
Symplectic group
Sp(n) Simple Lie groups List of simple Lie groups
Infinite simple Lie groups: A n B n ... n
Exceptional simple Lie groups: G F E E ...
Loop group
Lie algebras Exponential map Adjoint representation of a Lie group Adjoint representation of a Lie algebra Killing form ... Lie point symmetry Structure of semi-simple Lie groups Dynkin diagrams Cartan subalgebra Root system ... Representation of a Lie algebra Lie groups in Physics Particle physics and representation theory Representation theory of the Lorentz group Representation theory of the Poincaré group ... e In mathematics , a Lie algebra (pronounced /ˈliː/ ("lee"), not /ˈlaɪ/ ("lye")) 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" (after Sophus Lie ) was introduced by Hermann Weyl in the 1930s. In older texts, the name "

 2. Lie Algebra An algebra in which multiplication satisfies properties similar to the socalled bracket operation on matrices given by A, B = AB - BA, where the operation onhttp://www.daviddarling.info/encyclopedia/L/Lie_algebra.html

3. Lie Algebra - Definition
In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee ) is an algebraic structure whose main use lies in studying geometric objects such as
http://www.wordiq.com/definition/Lie_algebra
##### Lie algebra - Definition
In mathematics , a Lie algebra (named after Sophus Lie , pronounced "lee") is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds . Lie algebras were introduced to study the concept of infinitesimal transformation Contents showTocToggle("show","hide") 1 Definition
2 Category theoretic definition

3 Examples

4 Homomorphisms, subalgebras, and ideals
...
7 References
##### Definition
A Lie algebra is a type of an algebra over a field ; it is a vector space g over some field F together with a binary operation g g g , called the Lie bracket , which satisfies the following properties:
• Bilinearity:
for all a b F and all x y z g
• The Jacobi identity
for all x y z in g
• For all x in g
Note that the first and third properties together imply for all x y in g ("anti-symmetry"). Conversely, the antisymmetry property implies property 3 above as long as F is not of characteristic 2. Note also that the multiplication represented by the Lie bracket is not in general associative
##### Category theoretic definition
A Lie algebra is an object A in the category of vector spaces
Liealgebra.png

4. En (Lie Algebra) - Wikipedia, The Free Encyclopedia
E7� has dimension 190, but is not a simple Lie algebra it contains a 57
http://en.wikipedia.org/wiki/En_(Lie_algebra)
##### edit Finite dimensional Lie algebras
• E is another name for the Lie algebra A A of dimension 11. E is another name for the Lie algebra A of dimension 24. E is another name for the Lie algebra D of dimension 45. E is the exceptional Lie algebra of dimension 78. E is the exceptional Lie algebra of dimension 133. E is the exceptional Lie algebra of dimension 248.
##### edit Infinite dimensional Lie algebras
• E is another name for the infinite dimensional affine Lie algebra E (or E8 lattice ) corresponding to the Lie algebra of type E E is an infinite dimensional Kac–Moody algebra whose root lattice is the even Lorentzian unimodular lattice II of dimension 10. Some of its root multiplicities have been calculated; for small roots the multiplicities seem to be well behaved, but for larger roots the observed patterns break down. E is an infinite dimensional Kac–Moody algebra that has been conjectured to generate the symmetry "group" of M-theory E n for n ≥12 is an infinite dimensional Kac–Moody algebra that has not been studied much.

5. Quasi-Lie Algebra - Wikipedia, The Free Encyclopedia
In mathematics, a quasiLie algebra in abstract algebra is just like a Lie
http://en.wikipedia.org/wiki/Quasi-Lie_algebra
##### Quasi-Lie algebra
Please help improve this article by adding citations to reliable sources . Unsourced material may be challenged and removed (December 2009) In mathematics , a quasi-Lie algebra in abstract algebra is just like a Lie algebra , but with the usual axiom
x x
replaced by
x y y x (anti-symmetry).
In characteristic other than 2, these are equivalent (in the presence of bilinearity ), so this distinction doesn't arise when considering real or complex Lie algebras. It can however become important, when considering Lie algebras over the integers. In a quasi-Lie algebra,
x x
Therefore the bracket of any element with itself is 2-torsion, if it does not actually vanish.
Retrieved from " http://en.wikipedia.org/wiki/Quasi-Lie_algebra Categories Lie algebras Hidden categories: Articles lacking sources from December 2009 All articles lacking sources Personal tools Namespaces Variants Views Actions Search Navigation Interaction Toolbox Print/export

 6. Menu Page For /Murillo-Buijs/lie_algebra /MurilloBuijs/lie_algebra.first.dvi (6920 bytes). /Murillo-Buijs/lie_algebra. dvi (70357 bytes) /Murillo-Buijs/lie_algebra.abstract (506 bytes).http://hopf.math.purdue.edu/cgi-bin/generate?/Murillo-Buijs/lie_algebra

7. Lie Algebra - Slider
In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee ) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and
http://enc.slider.com/Enc/Lie_algebra
Encyclopaedia

L

Li

Lia
...
Liz
##### Lie algebra
In mathematics , a Lie algebra (named after Sophus Lie , pronounced "lee") is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds . Lie algebras were introduced to study the concept of infinitesimal transformations
##### Definition
A Lie algebra is a type of an algebra over a field ; it is a vector space g over some field F together with a binary operation g g g , called the Lie bracket , which satisfies the following properties: for all a b F and all x y z g
for all x y z in g Note that the first and second properties together imply for all x y in g ("anti-symmetry"). Conversely, the antisymmetry property implies property 2 above as long as F is not of characteristic Also note that the multiplication represented by the Lie bracket is not in general associative rings or associative algebras in the usual sense, although much of the same language is used to describe them.

 8. The Rational Homotopy Lie Algebra Of Function Spaces Urtzi Buijs File Format PDF/Adobe Acrobat Quick Viewhttp://hopf.math.purdue.edu/Murillo-Buijs/lie_algebra.pdf

9. Lie Algebra - Wiktionary
Nov 13, 2007 Lie algebra. Definition from Wiktionary, the free dictionary Retrieved from http//en.wiktionary.org/wiki/lie_algebra
http://en.wiktionary.org/wiki/Lie_algebra
##### Lie algebra
Definition from Wiktionary, the free dictionary Jump to: navigation search Wikipedia has an article on: Lie algebra Wikipedia
##### Contents

10. Lie Algebra: Facts, Discussion Forum, And Encyclopedia Article
Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from
http://www.absoluteastronomy.com/topics/Lie_algebra
Home Discussion Topics Dictionary ... Login Lie algebra
##### Lie algebra
Overview In mathematics Mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a Lie algebra ( ("lee"), not ("lye")) is an algebraic structure Algebraic structure In algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
whose main use is in studying geometric objects such as 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...
s and differentiable manifold Differentiable manifold A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
s. Lie algebras were introduced to study the concept of

11. Reductive Lie Algebra - Definition
In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee ) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and
http://www.wordiq.com/definition/Reductive_Lie_algebra
##### Reductive Lie algebra - Definition
In mathematics , a Lie algebra (named after Sophus Lie , pronounced "lee") is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds . Lie algebras were introduced to study the concept of infinitesimal transformation Contents showTocToggle("show","hide") 1 Definition
2 Category theoretic definition

3 Examples

4 Homomorphisms, subalgebras, and ideals
...
7 References
##### Definition
A Lie algebra is a type of an algebra over a field ; it is a vector space g over some field F together with a binary operation g g g , called the Lie bracket , which satisfies the following properties:
• Bilinearity:
for all a b F and all x y z g
• The Jacobi identity
for all x y z in g
• For all x in g
Note that the first and third properties together imply for all x y in g ("anti-symmetry"). Conversely, the antisymmetry property implies property 3 above as long as F is not of characteristic 2. Note also that the multiplication represented by the Lie bracket is not in general associative
##### Category theoretic definition
A Lie algebra is an object A in the category of vector spaces
Liealgebra.png

12. Lie Algebra/quadratic Casimir Invariant - Mathematics Wiki
Mar 17, 2009 For any simple Lie algebra, the quantity. t^2 \equiv t^a t^a\, ,. commutes with all the other elements of the algebra
##### From Mathematics wiki
Lie algebra Jump to: navigation search Here we assume a compact Lie group , so that we can write in some representation . For any simple Lie algebra , the quantity commutes with all the other elements of the algebra: , since is antisymmetric in its last two indices. This object is an invariant of the algebra, known as the quadratic Casimir invariant or quadratic Casimir operator or simply the quadratic Casimir . The (irreducible) matrix representation of is therefore proportional to the identity matrix (since commutes with all , and with , it commutes with all , and by Schur's lemma where labels the representation. Also, is sometimes labeled as In the adjoint representation, If are suitably normalized, so that is completely antisymmetric, then this can be written as Now, in a particular irreducible representation but also Therefore we obtain the formula
##### References
• An Introduction to Quantum Field Theory ISBN 978-0201503975

 13. Lie Algebra - VisWiki Lie algebra Lie group, Adjoint endomorphism, Killing form, Semisimple, Covering space - VisWikihttp://viswiki.com/en/Lie_algebra

14. Lie Algebra/su(n) - Mathematics Wiki
http://www.mathematics.thetangentbundle.net/wiki/Lie_algebra/su(n)
##### Algebra
Writing one finds that
Jacobi identity
It also follows that
##### Completeness
Suppose we write the inner product as Viewed as dimensional vectors in complex Euclidean space , this is just the Euclidean norm . We therefore have a subspace spanned by orthonormal vectors. If these were a complete set , then we would have the completeness relation summation implied ). However, since the generators are traceless , they are all orthogonal to , the dimensional vector corresponding to . For , so we simply project out the subspace In matrix form , this becomes (keeping in mind that the generators are also hermitian
Consider the tensor product of the antifundamental representation and fundamental representations of , i.e., . It transforms as Now the trace singlet ) part is invariant, so transforms as the . The trace-free part stays trace-free under a transformation owing to the cyclic property of the trace. The trace-free part of can be written as a linear combination of some basis of traceless matrices. There happen to be

 15. Lie Algebra | Ask.com Encyclopedia Definition and first properties. A Lie algebra is a vector space over some field F together with a binary operation , called the Lie bracket, which satisfies thehttp://www.ask.com/wiki/Lie_algebra?qsrc=3044

16. Lie Algebra
In mathematics, a Lie algebra (pronounced ( lee ), not ( lye )) is an algebraic structure whose main use is in studying geometric objects such as Lie groups
http://pediaview.com/openpedia/Lie_algebra
##### Lie algebra
Lie groups
Classical groups
General linear group GL(n)
Special linear group
SL(n)
Orthogonal group
O(n)
Special orthogonal group
SO(n)
Unitary group
U(n)
Special unitary group
SU(n)
Symplectic group
Sp(n) Simple Lie groups List of simple Lie groups
Infinite simple Lie groups: A n B n ... n
Exceptional simple Lie groups: G F E E ...
Loop group
Lie algebras Exponential map Adjoint representation of a Lie group Adjoint representation of a Lie algebra Killing form ... Lie point symmetry Structure of semi-simple Lie groups Dynkin diagrams Cartan subalgebra Root system ... Representation of a Lie algebra Lie groups in Physics Particle physics and representation theory Representation theory of the Lorentz group Representation theory of the Poincaré group ... Representation theory of the Galilean group d In mathematics , a Lie algebra (pronounced ("lee"), not ("lye")) 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" (after Sophus Lie ) was introduced by Hermann Weyl in the 1930s. In older texts, the name "

17. Lie Algebra/root - Mathematics Wiki
http://www.mathematics.thetangentbundle.net/wiki/Lie_algebra/root
##### From Mathematics wiki
Lie algebra Jump to: navigation search Let be a semisimple Lie algebra and consider its adjoint representation represented on a Hilbert space basis as Let be elements of the Cartan subalgebra of , which we may simultaneously diagonalize . Then the eigenvalues of are called the roots , which are simply the weights in the adjoint representation . The eigenvectors of the are the root vectors . We may write or It follows that , so that roots occur in pairs . Now raises the weight of a vector by , since so, has zero weight, and must lie in . Let . Then and Define and notice that which form an su(2) subalgebra Retrieved from " http://www.mathematics.thetangentbundle.net/wiki/Lie_algebra/root Views Personal tools Navigation Search Toolbox

 18. Lie Algebras On the other hand, ifgisthe Lie algebra ofa Lie group G, then there is an exponential map exp g! G, and this is what is meant by the exponentials on the left of (1.2).http://www.math.harvard.edu/~shlomo/docs/lie_algebras.pdf

 19. The Lie Algebra Of Infinitesimal Symmetries Of Nonlinear Diffusion File Format PDF/Adobe Acrobat Quick Viewhttp://doc.utwente.nl/60541/1/lie_algebra.pdf

20. Lie Algebra
Definition. A Lie algebra is a vector space g over some field F (typically the real or complex numbers) together with a binary operation , g g g, called the Lie
http://www.fact-index.com/l/li/lie_algebra_1.html
Main Page See live article Alphabetical index
##### Lie algebra
In mathematics , a Lie algebra (pronounced as "lee", named in honor of Sophus Lie ) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds Table of contents 1 Definition
2 Examples

3 Homomorphisms, Subalgebras and Ideals

4 Classification of Lie Algebras
##### Definition
A Lie algebra is a vector space g over some field F (typically the real or complex numbers) together with a binary operation g g g , called the Lie bracket , which satisfies the following properties:
• it is bilinear , i.e., [ a x b y z a x z b y z ] and [ z a x b y a z x b z y ] for all a b in F and all x y z in g it satisfies the Jacobi identity , i.e., [[ x y z z x y y z x ] = for all x y z in g x x ] = for all x in g
Note that the first and third properties together imply [ x y y x ] for all x y in g ("anti-symmetry"). Note also that the multiplication represented by the Lie bracket is not in general associative , that is, [[ x y z ] need not equal [ x y z
##### Examples
Every vector space becomes a (rather uninteresting) Lie algebra if we define the Lie bracket to be identically zero. Euclidean space R becomes a Lie algebra with the Lie bracket given by the cross-product of vectorss If an associative algebra A with multiplication * is given, it can be turned into a Lie algebra by defining [

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