1. Semi-Simple Lie Algebras And Their Representations Book for particle physicists by Robert N. Cahn. Published by BenjaminCummings in 1984. Chapters in PostScript. http://www-physics.lbl.gov/~rncahn/book.html

I have placed a postscript copy of my book Semi-Simple Lie Algebras and their Representations, published originally by Benjamin-Cummings in 1984, on this site the publisher has returned the rights to the book to me, you are invited to take a copy for yourself. Preface, Table of Contents, Bibliography, Index 1 Chapter 1 SU(2) Chapter 2 SU(3) Chapter 3 The Killing Form Chapter 4 The Structure of Simple Lie Algebras Chapter 5 A Little about Representations Chapter 6 More on the Structure of Simple Lie Algebras Chapter 7 Simple Roots and the Cartan Matrix Chapter 8 The Classical Lie Algebras Chapter 9 The Exceptional Lie Algebras Chapter 10 More on Representations Chapter 11 Casimir Operators and Freudenthal's Formula Chapter 12 The Weyl Group Chapter 13 The Dimension Formula Chapter 14 Reducing Product Representations Chapter 15 Subalgebras Chapter 16 Branching Rules

2. Lie Algebra -- From MathWorld lie algebra, A nonassociative algebra obeyed by objects such as the Lie bracket andPoisson bracket. Elements f, g, and h of a lie algebra satisfy, (1). http://mathworld.wolfram.com/LieAlgebra.html

Algebra Group Theory Lie Theory Lie Algebra Lie Algebra A nonassociative algebra obeyed by objects such as the Lie bracket and Poisson bracket . Elements f g , and h of a Lie algebra satisfy and (the Jacobi identity ). The relation implies For characteristic not equal to two, these two relations are equivalent. The binary operation of a Lie algebra is the bracket An associative algebra A with associative product xy can be made into a Lie algebra by the Lie product Every Lie algebra L is isomorphic to a subalgebra of some where the associative algebra A may be taken to be the linear operators over a vector space V (the ; Jacobson 1979, pp. 159-160). If L is finite dimensional, then V can be taken to be finite dimensional ( Ado's theorem for characteristic p Iwasawa's theorem for characteristic The classification of finite dimensional simple Lie algebras over an algebraically closed field of characteristic can be accomplished by (1) determining matrices called Cartan matrices corresponding to indecomposable simple systems of roots and (2) determining the simple algebras associated with these matrices (Jacobson 1979, p. 128). This is one of the major results in Lie algebra theory, and is frequently accomplished with the aid of diagrams called Dynkin diagrams Ado's Theorem Derivation Algebra Dynkin Diagram ... Weyl Group References Humphrey, J. E.

3. Lie Algebra - Home pages are devoted to a new way of viewing a lie algebra. The successes of such methods as symplectic geometry, Kirillov http://www.math.siu.edu/kocik/lie/lie-home.htm

What is a Lie algebra A new geometric view These pages are devoted to a new way of viewing a Lie algebra. The successes of such methods as symplectic geometry, Kirillov-Constant-Souriau quantization, Lax equations, etc., call for a more geometric treatment of the content of Lie algebras. Here is a proposition of a new approach within which many known objects are re-defined and new are introduced. Presented here are or will be: Lie algebra via tensors - with an interactive page. New differential-geometric objects on a Lie algebra! Euler's top and all that Programs coded in Maple Lie alg home Lie maps Lie mandala diff geo learn graphs homepage of JK

4. Quantum Lie Algebras Current status of research on Quantum lie algebras and bibliography. Includes Mathematica notebooks for calculations and animations of quantum root systems. http://www-users.york.ac.uk/~gwd2/q-lie.html

Unfortunately you are using a browser which can not handle frames. Therefore you will not be able to see the navigational tools and will also not be able to display the animated root system. Click to proceed

5. Lie Algebra Root -- From MathWorld Math Contributors , Rowland v. lie algebra Root, The roots of a semisimple Liealgebra are the lie algebra weights occurring in its adjoint representation. http://mathworld.wolfram.com/LieAlgebraRoot.html

Algebra Group Theory Lie Theory Lie Algebra ... Rowland Lie Algebra Root This entry contributed by Todd Rowland The roots of a semisimple Lie algebra are the Lie algebra weights occurring in its adjoint representation . The set of roots form the root system , and are completely determined by . It is possible to choose a set of Lie algebra positive roots , every root is either positive or is positive. The Lie algebra simple roots are the positive roots which cannot be written as a sum of positive roots. The simple roots can be considered as a linearly independent finite subset of Euclidean space , and they generate the root lattice . For example, in the special Lie algebra of two by two matrices with zero matrix trace , has a basis given by the matrices The adjoint representation is given by the brackets so there are two roots of given by and . The Lie algebraic rank of is one, and it has one positive root. Cartan Matrix Lie Algebra Lie Algebra Weight Semisimple Lie Algebra ... Weyl Group Author: Eric W. Weisstein Wolfram Research, Inc.

6. What IS A Lie Group? An example of a solvable Lie group is the nilpotent Lie group that can be formedfrom the nilpotent lie algebra of upper triangular NxN real matrices. http://www.innerx.net/personal/tsmith/Lie.html

Tony Smith's Home Page What IS a Lie Group? Thanks to John Baez and Dave Rusin for pointing out that this page is a non-rigorous, non-technical attempt at answering the question ONLY for compact real forms of complex simple Lie groups, such as groups of rotations acting on spheres, for which a complete classification is known. There are a lot of Lie groups that are NOT compact real forms of complex simple Lie groups. For instance, the real line with the action of translation is a non-compact Lie group, and solvable Lie groups are certainly not simple groups. An example of a solvable Lie group is the nilpotent Lie group that can be formed from the nilpotent Lie algebra of upper triangular NxN real matrices. So, when you read this page, be SURE to realize that when I say "Lie group", that is my shorthand for "compact real form of a complex simple Lie group", and similar shorthand is being used when I say "Lie algebra". As it will turn out that the Lie groups I will discuss are closely related to the division algebras, I will note that you can find a lot about the division algebras on Dave Rusin's division algebra fact page At the end of this page, some miscellaneous related matters are discussed:

7. Lie Algebra a basis of a lie algebra. Properties lie algebra vector space with dim = dim of corresponding group. lie algebra http://www-cgi.cs.cmu.edu/People/jcl/classnotes/math/group_theory/lie_algebra.ht

The generator s of a continuous group form a basis of a lie algebra . Properties: lie algebra vector space with dim = dim of corresponding group lie algebra closed under commutat ion [ , ]. [Xi,Xj]=ifijkXk where fijk = structure constants of the lie algebra For a compact Lie group fijk=-fjik always there exists a basis of generator s s.t. fijk=eijk= antisymmetric tensor Jacobi identity cyclic permutations)=0 Xi's are hermitian A lie algebra can correspond to multiple group s. Examples: SU (4) and SO SU (2) and SO SO (4) and SU (2)x SU SO (5) and SP source jl@crush.caltech.edu index ... GL

8. Lie Algebra E8 From Clifford Algebra Cl(8) Tony Smith's Home Page. E8 lie algebra from Clifford Algebra Cl(8). The 28dimensionalgrade-2 bivectors of Cl(8) form the lie algebra Spin(8). http://www.innerx.net/personal/tsmith/E8fromClif8.html

Tony Smith's Home Page Lie Algebra from Clifford Algebra Cl(8) and also of my VoDou Physics Model , as well as and The Clifford Algebra Cl(8) has dimension 2^8 = 256. Since 256 = 16 x 16 = 2^4 x 2^4, the full spinors of Cl(8) are 16-dimensional, and the half-spinors of Cl(8) are 8-dimensional Cl(8) has graded structure The 28-dimensional grade-2 bivectors of Cl(8) form the Lie Algebra Spin(8) The E8 Lie Algebra is the sum (on this page I am using the word "sum" very imprecisely) of the 120-dimensional Spin(16) Lie Algebra of the 2^16 = 256 x 256 = 65,536-dimensional Cl(16) Clifford Algebra and one 2^7 = 128-dimensional Cl(16) half-spinor space To construct E8 from Cl(8): First , construct the 120-dimensional Spin(16) from Cl(8): Since Cl(16) can be written as the tensor product Cl(16) = Cl(8) x Cl(8), the graded structure of Cl(16) can be written in terms of the graded structure of Cl(8) as follows: Therefore: Spin(16) = 120 = 28x1 + 8x8 + 1x28 = 28 + 64 + 28, and Spin(16) is the sum of two copies of Spin(8) plus the square of the 8-dimensional Cl(8) grade-1 vector space Second , construct the 128-dimensional half-spinors of Cl(16) from Cl(8): The Cl(8) half-spinors are 8-dimensional, so that the Cl(8) full-spinor space is 8e + 8o = 16-dimensional, where 8e is one half-spinor 8-dimensional space and 8o is the other mirror image half-spinor 8-dimensional space.

9. Midatl.html Topics DownUp Algebras; Extended Affine lie algebras. Virginia Tech, Blacksburg; 1011 March 2001. http://www.math.vt.edu/people/farkas/midatl.html

MID-ATLANTIC ALGEBRA CONFERENCE Virginia Tech Blacksburg, Virginia Saturday March 10, 2001 Sunday March 11, 2001 PRINCIPAL SPEAKER: Professor Georgia Benkart University of Wisconsin Down-Up Algebras Extended Affine Lie Algebras Click here for a biography of our speaker and a description of the two lectures. A final schedule is now posted. Reminder: accomodations are at the Microtel Inn. Visit here for directions to the motel and campus. If you have questions, please contact Dan Farkas Department of Mathematics Virginia Tech 24061-0123 farkas@math.vt.edu

10. LECTURES ON INFINITE-DIMENSIONAL LIE ALGEBRA LECTURES ON INFINITEDIMENSIONAL lie algebra by Minoru Wakimoto (Kyushu University,Japan) The representation theory of affine lie algebras has been developed http://www.wspc.com/books/mathematics/4269.html

Home Browse by Subject Bestsellers New Titles ... Browse all Subjects Search Keyword Author Concept ISBN Series New Titles Editor's Choice Bestsellers Book Series ... Join Our Mailing List LECTURES ON INFINITE-DIMENSIONAL LIE ALGEBRA by Minoru Wakimoto (Kyushu University, Japan) The representation theory of affine Lie algebras has been developed in close connection with various areas of mathematics and mathematical physics in the last two decades. There are three excellent books on it, written by Victor G Kac. This book begins with a survey and review of the material treated in Kac's books. In particular, modular invariance and conformal invariance are explained in more detail. The book then goes further, dealing with some of the recent topics involving the representation theory of affine Lie algebras. Since these topics are important not only in themselves but also in their application to some areas of mathematics and mathematical physics, the book expounds them with examples and detailed calculations. Contents: Preliminaries on Affine Lie Algebras Characters of Integrable Representations Principal Admissible Weights Residue of Principal Admissible Characters Characters of Affine Orbifolds Operator Calculus Branching Functions W-Algebra Vertex Representations for Affine Lie Algebras Soliton Equations Readership: Graduate students and researchers interested in representation theory, combinatorics, vertex algebras, modular forms, soliton equations, particle physics and solvable models.

11. The Algebra Group Of The LUC LUC Algebra Group. Major areas of research include Noncommutative geometry; Invariant theory; Group algebras and Schur algebras; lie algebra; Maximal orders. http://alpha.luc.ac.be/Research/Algebra/

The Algebra Group of the LUC This is the Home Page of the Algebra Group at the Limburgs Universitair Centrum What's new Major areas of research include: Non-commutative geometry Invariant theory Group algebras and Schur algebras Lie algebra Maximal orders Members of the group: Prof. Dr. Alfons Ooms Prof. Dr. Erna Nauwelaerts Prof. Dr. Paul Wauters Prof. Dr. Michel Van den Bergh Prof. Dr. Lydia Delvaux Kristien Bauwens ... Sevenhant Home Pages of other mathematicians Click Here Mathematical News Groups Click Here Other mathematics departments in Belgium Katholieke Universiteit Leuven Campus Kortrijk Katholieke Universiteit Leuven University of Antwerp Université Catholique de Louvain ... Vrije Universiteit Brussel Author : Michel Van den Bergh

12. When Is A Lie Algebra Not A Lie Algebra? When is a lie algebra not a lie algebra? We look at weight systems on Feynman diagrams. A metric lie algebra gives one example of a weight system. http://www.maths.ox.ac.uk/~sawon/lie_alg_obj.shtml

13. Lie Algebra - Wikipedia lie algebra. Examples. Every vector space becomes a (rather uninteresting)lie algebra if we define the Lie bracket to be identically zero. http://www.wikipedia.org/wiki/Lie_algebra

Main Page Recent changes Edit this page Older versions Special pages Set my user preferences My watchlist Recently updated pages Upload image files Image list Registered users Site statistics Random article Orphaned articles Orphaned images Popular articles Most wanted articles Short articles Long articles Newly created articles All pages by title Blocked IP addresses Maintenance page External book sources Printable version Talk Log in Help Lie algebra From Wikipedia, the free encyclopedia. A Lie algebra (pronounced as "lee", named in honor of Sophus Lie ) is an algebraic structure in mathematics whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds 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 property 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

14. Lie Algebra lie algebra. see also lie algebra , Lie Groups. Jacobson, Nathan. LieAlgebras. New York Dover, 1979. 331 p. $7.95. Mikhalev, Alexander http://www.ericweisstein.com/encyclopedias/books/LieAlgebra.html

Lie Algebra see also Lie Algebra Lie Groups Jacobson, Nathan. Lie Algebras. New York: Dover, 1979. 331 p. $7.95. Mikhalev, Alexander A. and Zolotykh, Anrej A. Combinatorial Aspects of Lie Superalgebras. Boca Raton, FL: CRC Press, 1955. 272 p. $115. http://www.ericweisstein.com/encyclopedias/books/books/LieAlgebra.html

15. A Monster Lie Algebra? R. E. Borc Herds, J. H. Con W A Y , L. Queen A Monster lie algebra? R. E. Borcherds, J. H. Conway, L. Queen and N. J. A. Sloane A version of this paper was originally published in Advances in Mathematics, vol. (1984), no. 1, pp. 7579. http://www.research.att.com/~njas/doc/bcqs.pdf

16. Victor Ginzburg - Principal Nilpotent Pairs In A Semisimple Lie Algebra of pairs of commuting nilpotent elements in a semisimple lie algebra. These pairs enjoy quite remarkable properties and http://www.cms.math.ca/Events/winter99/abstracts/node59.html

home about the CMS media releases search ... other societies Next: Mark Haiman - The Up: Algebraic Combinatorics, Group Representations Previous: Tudose Geanina - Littlewood-Richardson Victor Ginzburg - Principal nilpotent pairs in a semisimple Lie algebra VICTOR GINZBURG, University of Chicago Principal nilpotent pairs in a semisimple Lie algebra We introduce and study a new class of pairs of commuting nilpotent elements in a semisimple Lie algebra. These pairs enjoy quite remarkable properties and are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue of the Kostant partition function, and propose the corresponding two-parameter analogue of the weight multiplicity formula. In a different direction, each principal nilpotent pair gives rise to a harmonic polynomial on the Cartesian square of the Cartan subalgebra, that transforms under an irreducible representation of the Weyl group. In the special case of GL n , the conjugacy classes of principal nilpotent pairs and the irreducible representations of the Symmetric group, S n , are both parametrised (in a compatible way) by Young diagrams. In general, our theory provides a natural generalization to arbitrary Weyl groups of the classical construction of simple

17. Maps Of Lie Algebra What is a lie algebra, really. A geometric view. These maps can be put intoone diagram graph of a Lie map This is the mandala of a lie algebra. http://www.math.siu.edu/kocik/lie/lie-map.htm

What is a Lie algebra, really A geometric view The standard view: a linear space L with a product, i.e., a bilinear map L x L -> L denoted a,b > [a,b] Let us however view it as a linear space L with a (1,2)-variant tensor c. We may view c as a map L* x L x L > R Different restrictions of this map to a fewer number of arguments result in major concepts of Lie agebra: These maps can be put into one diagram: This is the mandala of a Lie algebra. Click here to get to an interactive page, where you can click on a particular item of the mandala to get further definitions. You may see the full-size version a of this picture (you will need to scroll the screen). For the printing purposes download the Lie map in jpeg format 66 Kb Lie alg home Lie maps Lie mandala diff geo learn graphs HomePage of JK

18. Weyl Groups AD-E Series of lie algebras the root vectors of the 24+4 = 28-dimensional D4 lie algebra; two 4-dimensional HyperOctahedra, lying (in a 5th http://www.innerx.net/personal/tsmith/Weyl.html

Tony Smith's Home Page A D E Series of Lie Algebras: AN for Large N and A-D-E DN for Large N Physical interpretations are motivated by Saul-Paul Sirag 's ideas about A-D-E , and Weyl groups Del Pezzo Surfaces also correspond to the A-D-E series, as does the McKay Correspondence The root vector lattices of the E6-E7-E8 chain of Lie algebras are related to aperiodic Tilings in 2, 3, and 4 dimensions which can be thought of as Irrational Slices of an 8-dimensional E8 Lattice and its sublattices, such as . Here is one way to visualize the 240 vertices around the origin of the E8 lattice: The 2-dimensional Penrose Tiling in the above image was generated by Quasitiler as a section of a 5-dimensional cubic lattice based on the 5-dimensional HyperCube shown in the center above the Penrose Tiling plane. Here is a web page with a larger (548k gif) version of the above image The above-plane geometric structures in the above image are, going from left to right: 4-dimensional 24-cell , whose 24 vertices are the root vectors of the 24+4 = 28-dimensional D4 Lie algebra two 4-dimensional HyperOctahedra, lying (in a 5th dimension) above and below the 24-cell, whose 8+8 = 16 vertices add to the 24 D4 root vectors to make up the 40 root vectors of the 40+5 = 45-dimensional

19. PlanetMath: Lie Algebra lie algebra, (Definition). A lie algebra over a field is a vectorspace with a bilinear map , called the Lie bracket and denoted . http://planetmath.org/encyclopedia/LieAlgebra.html

Math for the people, by the people. Encyclopedia Books Papers Expositions ... Random Login create new user name: pass: forget your password? Main Menu the math Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Corrections talkback Polls Forums Feedback Bug Reports information Docs Classification News Legalese ... TODO List Lie algebra (Definition) A Lie algebra over a field is a vector space with a bilinear map , called the Lie bracket and denoted . It is required to satisfy for all The Jacobi identity for all A vector subspace of the Lie algebra is a subalgebra if is closed under the Lie bracket operation, or, equivalently, if itself is a Lie algebra under the same bracket operation as . An ideal of is a subspace for which whenever either or . Note that every ideal is also a subalgebra. Some general examples of subalgebras: The center of , defined by for all . It is an ideal of The normalizer of a subalgebra is the set . The Jacobi identity guarantees that is always a subalgebra of The centralizer of a subset is the set . Again, the Jacobi identity implies that

20. Lie Algebra Research lie algebra research. My main area of research is in the wonderfulworld of lie algebras, where I mostly inhabit the outer fringes http://it.stlawu.edu/~dmelvill/17b/

Lie algebra research My main area of research is in the wonderful world of Lie algebras , where I mostly inhabit the outer fringes of Kac-Moody algebras and some of their variants. Some of the topics I am interested in are: Borcherds algebras; Crystal base; PBW theorems; Root multiplicities; Superalgebras; Verma-type modules. If you find anything on this site interesting, or have any information or (p)reprints to share, please let me know. Go to Duncan J. Melville Last modified: 26 July 1996 Duncan J. Melville Comments to dmel@music.stlawu.edu

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