21. Lie Algebra http://www.win.tue.nl/~amc/oz/la.html

22. A Characterization Of The Lie Algebra Rank Condition By Transverse Periodic Func A Characterization of the lie algebra Rank Condition by Transverse Periodic Functions. Keywords. controllability, driftless system, transversality, lie algebra. http://epubs.siam.org/sam-bin/dbq/article/36605

SIAM Journal on Control and Optimization Volume 40, Number 4 pp. 1227-1249 A Characterization of the Lie Algebra Rank Condition by Transverse Periodic Functions Pascal Morin, Claude Samson Abstract. The Lie algebra rank condition plays a central role in nonlinear systems control theory. The present paper establishes that the satisfaction of this condition by a set of smooth control vector fields is equivalent to the existence of smooth transverse periodic functions. The proof here enclosed is constructive and provides an explicit method for the synthesis of such functions. Key words. controllability, driftless system, transversality, Lie algebra AMS Subject Classifications PII Retrieve PostScript document ( 36605.ps : 592622 bytes) Retrieve GNU Compressed PostScript document ( ... Retrieve reference links For additional information contact service@siam.org

23. The Free Lie Algebra The Free lie algebra. A good reference for the material in this section is 11. Alie algebra is a vector space L over k which is equipped with a bilinear map http://www.bangor.ac.uk/~mas019/symb/node2.html

Next: Data Structures and Algorithms Up: Notes on Symbolic Computation Previous: Introduction The Free Lie Algebra A good reference for the material in this section is [ ]. For simplicity, it is assumed that k is a field of characteristic not equal to in this section. A Lie algebra is a vector space L over k which is equipped with a bilinear map L L L which satisfies the following two axioms x y y x ] (skew-symmetry) and x y z y z x z x y ]] = (Jacobi identity) There is a free Lie algebra generated by a given set of symbols X . Roughly, it is the Lie algebra which satisfies the axioms above and no other relations. It is known that the free Lie algebra generated by a set X has a graded basis (every element in the basis has a non-negative integer called the degree associated to it) in a well defined way. If the cardinality of X is the finite number g , the number of basis elements of degree exactly c , for c is given by the recursive formula: l g l g g l g c m l g m where m c means that m divides c P. Hall (see [ ]) gave an algorithm which can be used to construct a graded basis H H c for the free Lie algebra F g generated by any set X x x g and any choice of a total order on X (which is assumed to be the obvious one indicated here). The construction is by recursion and an ordering on the basis is also constructed along the way. This is done in a way that not only are all the

24. Data Structures And Algorithms For The Free Lie Algebra The Free lie algebra Data Structures and Algorithms for the Free LieAlgebra. The free lie algebra on g generators was implemented http://www.bangor.ac.uk/~mas019/symb/node3.html

Next: Non Abelian Rewriting and Up: The Free Lie Algebra Previous: The Free Lie Algebra Data Structures and Algorithms for the Free Lie Algebra The free Lie algebra on g generators was implemented in the AXIOM system [ ] by the author in . The construction will be reviewed here. The Hall process builds up iterated products of a special nature in the free non-associative algebra. Instead of the usual notation xy for a product of two elements, the bracket notation x y is used and instead of calling x y a product, the term commutator is used. The Hall process works by building up a set of commutators and a total order on them inductively. These special elements are called basic commutators and the degree of a basic commutator is often called its weight A good data structure for basic commutators is a structure with three non-negative integer fields lft wt and rt . The wt field is needed for the implementation of the Hall algorithm below. There is a function from pairs of non-negative integers to non-negative integers called index . The value of index lft rt is the unique positive integer that gives the position of the basic commutator represented by the structure lft wt rt , mentioned above, in the total order given by the Hall construction (it is also constructed by the recursive procedure) and the value is zero if the structure is not in the basic commutator list.

25. Basic Lie Algebra Calculations With MAPLE Basic lie algebra calculations with MAPLE. To find the ith basis element ofthe lie algebra of type A_n, use the Liebasiselement_A(n,i) command. http://web.usna.navy.mil/~wdj/dynkin0.htm

Basic Lie algebra calculations with MAPLE This worksheet assumes you have MAPLE V5, with its share library. If not, you will need to read in the files coxeter.mpl from John Stembridge's site and dynkin.mpl The share package dynkin contains routines to do basic matrix computations with classical Lie algebras (G_2 is also available in a more recent version) and some plotting of weight spaces. restart;with(share):with(plots):with(coxeter);with(weyl); with(dynkin); The MAPLE V5 share package crystal may have trouble loading correctly on some machines. If so, you can download the crystal25.mpl file from the crystal page and load it using the read command. read(`d:/maplestuff/crystal/crystal25.mpl`); The Dynkin diagram and extended Dynkin diagrams of any simple Lie algebra are obtained by using the commands dynkin[dynkin_diagram] and dynkin[ex_dynkin_diagram]. The package weyl tells us a basis for the weights of C3 using the weights command. The package crystal (which uses the negative of the weights from the weyl package) tells us the weights occuring in the representation R of C3 with highest weight e3 using the command crystal[weight_system] . The weights occurring in the output of this are expresses as vectors in the vector space spanned by the weight space. The "crystal graph" of this representation is the labeled graph whose vertices are the weights occuring in R and two vertices are connected by an edge if a root vector sends the one weight space to the other (the edge will then be labeled by the corresponding root). To obtain this graph, you must create the graph using the

26. Finite Dimensional Representations Of Lie Algebras Let L denote a semisimple lie algebra over the complex numbers with a fixed Cartansubalgebra H. We may assume that L is generated (as a lie algebra) by a http://web.usna.navy.mil/~wdj/crystal2.htm

Finite dimensional representations of Lie algebras - a brief introduction using using crystal graphs a -a a are the corresponding root vectors. Let (s,V) denote an irreducible finite dimensional representation of L, s : L > End(V). The restriction of (s,V) to the Cartan subalgebra H decomposes into a direct sum of 1-dimensional representations of H. These are the weight spaces of (s,V). We shall need to assume that this decomposition is multiplicity free. This representation is uniquely determined by the action of the simple root vectors on the lowest weight vector of V. For example, the first several highest weight vectors of the Lie algebra A and the dimensions of the corresponding irreducible representations looks like: weight system of A We associate to (s,V) a digraph G as follows: the vertices of G are indexed by the weight spaces of (s,V) and we draw an edge from vertex W to vertex W' labeled with X a (a in D) if X a (W)=W'. This is the crystal graph of (s,V). Example: Let L= B and let (s,V) denote the standard 11-dimensional fundamental representation associated to the weight e5 (in the notation of Stembridge 's coxeter package). The crystal graph looks like:

27. The Type Of A Semisimple Lie Algebra NextPrev Right Left Up Index Root The Type of a Semisimple lie algebra.SemiSimpleType(L) AlgLie AlgLie. Let L be a lie algebra. http://magma.maths.usyd.edu.au/magma/htmlhelp/text953.htm

[Next] [Prev] [Right] [Left] ... [Root] The Type of a Semisimple Lie Algebra Let L be a Lie algebra. If L has a nondegenerate Killing form, then (over some algebraic extension of the ground field) L is the direct sum of absolutely simple Lie algebras. These Lie algebras have been classified and the classes are named A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4 and G_2. This function returns a single string containing the types of the direct summands of L. Example We compute the semisimple type of the Levi subalgebra of the simple Lie algebra of type D_7. [Next] [Prev] [Right] [Left] ... [Root]

28. Re: Lie Algebra Cohomology IndexThread Index Re lie algebra cohomology. Subject Re Liealgebra cohomology; From Aaron Bergman abergman@Princeton.EDU ; http://www.lns.cornell.edu/spr/2001-03/msg0031518.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Re: Lie algebra cohomology Subject : Re: Lie algebra cohomology From Date : 1 Mar 2001 23:00:41 GMT Approved : spr@rosencrantz.stcloudstate.edu (sci.physics.research) Newsgroups : sci.physics.research Organization : Princeton University References 3A9C29E7.2D760DDF@math.columbia.edu User-Agent : MT-NewsWatcher/3.1 (PPC) 3A9C29E7.2D760DDF@math.columbia.edu http://www.princeton.edu/~abergman/ References Re: Lie algebra cohomology From: Prev by Date: Re: Quantum Gravity Seminar: Week 12: Track 1 Next by Date: Re: Supersymmetry question Prev by thread: Re: Lie algebra cohomology Next by thread: Re: Lie algebra cohomology Index(es): Date Thread

29. Re: Lie Algebra Cohomology Re lie algebra cohomology. Subject Re lie algebra cohomology; FromPeter Woit woit@math.columbia.edu ; Date Thu, 1 Mar 2001 015449 GMT; http://www.lns.cornell.edu/spr/2001-03/msg0031475.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Re: Lie algebra cohomology Subject : Re: Lie algebra cohomology From Date : Thu, 1 Mar 2001 01:54:49 GMT Approved : mmcirvin@world.std.com (sci.physics.research) Newsgroups : sci.physics.research Organization : Columbia University References Sender : mmcirvin@world.std.com (Matthew J McIrvin) Follow-Ups Re: Lie algebra cohomology From: "A.J. Tolland" <ajt@hep.uchicago.edu> Re: Lie algebra cohomology From: Re: Lie algebra cohomology From: shocklee@phoenix.Princeton.EDU (Paul D. Shocklee) Prev by Date: Re: Quantum Gravity Seminar Next by Date: Re: WWW - New Space News Site Prev by thread: Re: Quantum Gravity Seminar Next by thread: Re: Lie algebra cohomology Index(es): Date Thread

30. Free Lie Algebra N. Bergeron, F. Bergeron and A. Garsia, Idempotents for the Free lie algebra andqEnumeration, Springer-Verlag, IMA Volumes in Mathematics, 19 (1989), 166-190 http://msfac1.math.yorku.ca/~nantel/Web/MoreWeb/loginsrv/lie.html

Publications N. Bergeron, F. Bergeron and A. Garsia Idempotents for the Free Lie Algebra and q-Enumeration Springer-Verlag, IMA Volumes in Mathematics, 19 (1989), 166-190. Minor Contribution in the Following Papers A. Garsia Combinatorics of the Free Lie Algebra and the Symmetric Group Analysis, Et Cetera, Research papers published in honor of Jurgen Moser's 60th birthday, ed: Paul H. Rabinowitz and Eduard Zehnder, Academic Press, 1990. See also Complete List symmetric functions and group representations Coxeter arrangement of hyperplanes descent algebras of Coxeter groups ... knot theory

31. Computing The Derivation Lie Algebra Of The Quadratic Jordan Algebra H3(Os,-) At Computing the derivation lie algebra of the quadratic Jordan algebraH3(Os,) at any characteristic. We have implemented the quadratic http://math1.uibk.ac.at/mathematik/jordan/archive/imH3split/

Computing the derivation Lie algebra of the quadratic Jordan algebra H3(Os,-) at any characteristic. We have implemented the quadratic Jordan structure of H3(Os,-), where Os denotes the split octonions, into a computer system (Mathematica). We have used this implementation to compute a generic expression of any element of f4(Os) at any characteristic. imH3split.pdf P. Alberca Bjerregaard

32. Liealg_overview.html Introduction to the lie algebra Package. by Yuly Billig (billig@math.carleton.ca)and Matthias Mazzag (m.mazzag@unb.ca). Free lie algebra. http://www.mapleapps.com/categories/mathematics/algebra/html/liealg_overview.htm

33. Dynkin.html Abstract Routines for drawing the Dynkin diagram of a simple lie algebra,plot weight systems and to create basis matrices structure constants. http://www.mapleapps.com/categories/mathematics/algebra/html/dynkin.html

34. RR-3873 : A Characterization Of The Lie Algebra Rank Condition By Transverse Per Translate this page RR-3873 - A characterization of the lie algebra Rank Condition by transverse periodicFunctions. KEY-WORDS NONLINEAR SYSTEM / HOMOGENEOUS SYSTEM / lie algebra. http://www.inria.fr/rrrt/rr-3873.html

RR-3873 - A characterization of the Lie Algebra Rank Condition by transverse periodic Functions Morin, Pascal Samson, Claude Les rapports de cet auteur Rapport de recherche de l'INRIA- Sophia Antipolis Page d'accueil de l'unité de recherche Fichier PostScript / PostScript file Fichier postscript du document : 183 Ko Fichier PDF / PDF file Fichier PDF du document : 334 Ko Projet : ICARE - 27 pages - Janvier 2000 - Document en anglais Page d'accueil du projet Abstract : The Lie Algebra Rank Condition (LARC) plays a central role in nonlinear systems control theory. The present paper establishes that the satisfaction of this condition by a set of smooth control vector fields is equivalent to the existence of smooth transverse periodic functions. The proof here enclosed is constructive and provides an explicit method for the synthesis of such functions. KEY-WORDS : NONLINEAR SYSTEM / HOMOGENEOUS SYSTEM / LIE ALGEBRA MOTS-CLES : SYSTEME NON LINEAIRE / SYSTEME HOMOGENE / ALGEBRE DE LIE

35. Topics In Lie Algebra Theory 2002/03 Giovanni Felder. Topics in lie algebra Theory. lie algebra cohomology andapplications. Winter Semester 2002/2003 Mondays, 13.1515.00, HG D1.2. http://www.math.ethz.ch/~felder/Lie0203/

Giovanni Felder Topics in Lie Algebra Theory Lie algebra cohomology and applications Winter Semester 2002/2003 Mondays, 13.15-15.00, HG D1.2 Announcement ( PS PDF Problem set 1, due 11 November [ PS PDF Problem set 2, due 18 November [ PS PDF Problem set 3, due 25 November [ PS PDF Problem set 4, due 9 December [ PS PDF Problem set 5, due 13 January [ PS PDF Problem set 6, due 27 January [ PS PDF Giovanni Felder Last modified: Mon Jan 20 10:40:44 CET 2003

36. Laplace Operators For The Lie Algebra Of Skew Symmetric Matrices Laplace Operators For The lie algebra Of Skew Symmetric Matrices. AlexanderI. Molev. Abstract Several constructions of Laplace operators http://wwwmaths.anu.edu.au/research.reports/mrr/95.042/abs.html

Mathematics Research Report MRR95-042 Laplace Operators For The Lie Algebra Of Skew Symmetric Matrices Alexander I. Molev Abstract: Several constructions of Laplace operators for the Lie algebra of skew-symmetric matrices are discussed. All of them are related with a Capelli-type determinant of the matrix formed by the generators of the Lie algebra. This service is maintained by the Mathematical Sciences Institute (MSI) Comments to webmaster@maths.anu.edu.au URL : http://wwwmaths.anu.edu.au/

37. [ref] 60.5 Properties Of A Lie Algebra 60.5 Properties of a lie algebra. IsLieAbelian( L ) P is true if L is a lie algebrasuch that each product of elements in L is zero, and false otherwise. http://wwwmaths.anu.edu.au/research.groups/algebra/GAP/www/Manual4/ref/C060S005.

60.5 Properties of a Lie Algebra IsLieAbelian( L ) P is true if L is a Lie algebra such that each product of elements in L is zero, and false otherwise. IsLieNilpotent( L ) P A Lie algebra L is defined to be (Lie) it nilpotent when its (Lie) lower central series reaches the trivial subalgebra. IsLieSolvable( L ) P A Lie algebra L is defined to be (Lie) it solvable when its (Lie) derived series reaches the trivial subalgebra. Top Previous Up Next ... Index GAP 4 manual February 2000

38. Atlas: The Lie Algebra Of Curves On Surfaces By Bill Goldman Abstracts Conference Homepage. The lie algebra of curves on surfacespresented by Bill Goldman University of Maryland This talk http://atlas-conferences.com/c/a/h/j/09.htm

Atlas Document # cahj-09 KNOTS in WASHINGTON XII Conference on Knot Theory and its Ramifications May 10-12, 2001 George Washington University Washington DC, USA Conference Organizers Dubravko Ivansic, Ilya Kofman, Jozef H.Przytycki, Yongwu Rong and Akira Yasuhara View Abstracts Conference Homepage The Lie algebra of curves on surfaces presented by Bill Goldman University of Maryland This talk will expound the Lie bialgebra based on oriented curves on an oriented surface. This Lie algebra was originally defined because of its representation in the Poisson algebra of the GL(n)-character variety of a compact surface. The Lie cobracket was discovered by Turaev. I will survey recent results on this Lie algebra and in particular discuss the relationship of acommutativity to disjointness. http://www.math.umd.edu/~wmg Date received: May 8, 2001 The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc.

39. Atlas: Analytic Joint Spectral Radius In A Solvable Lie Algebra Of Operators By Analytic joint spectral radius in a solvable lie algebra of operators presentedby Daniel Beltita Institute of Mathematics of the Romanian Academy http://atlas-conferences.com/c/a/e/o/12.htm

Atlas Document # caeo-12 18th International Conference on Operator Theory June 27 - July 1, 2000 University of the West Timisoara, Romania Conference Organizers Dumitru Gaspar, Traian Ceausu, Aurelian Craciunescu, Aurelian Gheondea, Radu-Nicolae Gologan, Ciprian Pop, Dan Popovici, Nicolae Suciu, Alexandru Terescenco, Dan Timotin and Flavius Turcu View Abstracts Conference Homepage Analytic joint spectral radius in a solvable Lie algebra of operators presented by Daniel Beltita Institute of Mathematics of the Romanian Academy The algebraic joint spectral radius of a set of operators (introduced by G.-C. Rota and G. Strang) has proved recently its usefulness in the solution of an important open problem concerning the existence of invariant subspaces for semigroups of compact quasinilpotent operators, cf. the paper of Yu.V. Turovskii in J. Funct. Anal. 162(1999), 313-322. On the other hand, several mathematicians have studied a geometric Our aim here is to introduce the concept of analytic spectral radius for a family of operators indexed by some finite measure space. This spectral radius is compared with the algebraic and geometric ones when the operators belong to some finite-dimensional solvable Lie algebra (extending the framework of commutativity by means of the recently introduced Cartan-Taylor joint spectrum). We describe several situations when the three spectral radii coincide. These results extend some of the corresponding facts concerning commuting n-tuples of operators.

40. What Is A Lie Algebra? What is a lie algebra? In early well. A lie algebra is a special exampleof a set with a function that satisfies different laws. These http://www.math.rutgers.edu/~nacin/def1.html

What is a Lie Algebra? In early schooling, concepts are taught such as multiplication and division. One first is taught a few examples, such as what 2 + 4 is. At some point they realize 2 + 4 = 4 + 2 and 3 + 6 = 6 + 3 and that this sort of symmetry works for any numbers. In general a + b = b + a for any a and b. This is called the commutative law. Notice that it also holds for multiplication as ab=ba for any real numbers a and b. Another important law is the associative law: (a + b) + c = a + (b + c). This one holds for multiplication as well, since (ab)c=a(bc) for any real numbers a, b, and c. What is addition really? We all have an instinctive feel for the concept. Basically addition takes two numbers, and gives back a third. Thus it is really a function, a sort of machine. We put in any two elements, which we take from the set (or collection) of numbers and we get a third. This third output number satisfies certain laws like the commutative and associative law above. Addition and multiplication satisfy many similar rules. One can step aside from these two examples of objects satisfying these rules and study all such objects which satisfy them. Taking the commutative law, associative law, and two other laws, we get a particular structure we call an abelian group. We can start to look for all the different examples of abelian groups instead of just individual examples. We can do the same for different algebraic structures as well.

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