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Topic: Dynkin diagram

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	root system (Site not responding. Last check: 2007-10-21)
		The Dynkin diagram can be extracted from the root system by choosing a base, that is a subset Δ of Φ which is a basis of V with the special property that every vector in Φ when written in the basis Δ has either all coefficients ≥0 or else all ≤0.
		The vertices of the Dynkin diagram correspond to vectors in Δ.
		Dynkin diagrams encode the inner product on E in terms of the basis Δ, and the condition that this inner product must be positive definite turns out to be all that is needed to get the desired classification.
	www.yourencyclopedia.net /root_system.html (1075 words)

	PlanetMath: Dynkin diagram
		Dynkin diagrams are a combinatorial way of representing the imformation in a root system.
		Thus Dynkin diagrams are finite graphs, with single, double or triple edges.
		This is version 1 of Dynkin diagram, born on 2003-02-13.
	planetmath.org /encyclopedia/DynkinDiagram.html (253 words)

	Root system - Wikipedia, the free encyclopedia
		Since Lie groups (and some analogues such as algebraic groups) became used in most parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied.
		To every root system is associated a graph (possibly with a specially marked edge) called the Dynkin diagram which is unique up to isomorphism.
		Given two root systems with the same Dynkin diagram, we can match up roots, starting with the roots in the base, and show that the systems are in fact the same.
	en.wikipedia.org /wiki/Root_system (1241 words)

	Dynkin (Site not responding. Last check: 2007-10-21)
		In 1948 Dynkin was awarded his Ph.D. and he became an assistant professor of Kolmogorov's who held the Probability Chair.
		Dynkin became Doctor of Physics and Mathematics in 1951 and Kolmogorov pressed for Dynkin to be awarded a chair.
		In 1977 Dynkin was appointed to Cornell University in Ithaca, New York.
	www-history.mcs.st-and.ac.uk /history/Mathematicians/Dynkin.html (865 words)

	Material to Scientific Biography
		Dynkin's most famous contribution to the theory of Lie algebras was his use of the "Coxeter-Dynkin diagrams" to describe and classify the Cartan matrices of semisimple Lie algebras.
		Dynkin's most famous contribution to the theory of Lie groups is, of course, the introduction and use of the notions of simple roots and the corresponding diagrams.
		Dynkin discovered the possibility of a purely measure-theoretic approach to this problem, under which the extremal measures are interpreted as measures with trivial behaviour of the trajectories at infinity.
	www.math.cornell.edu /~ebd/ebda.html (9079 words)

	[No title]
		Way back in "week62" I showed how a Dynkin diagram gives a finite reflection group: that is, a finite group of symmetries of n-dimensional Euclidean space, generated by reflections, one for each of the n dots in the diagram, satisfying relations described by the edges in the diagram.
		For example, I've already said this Dynkin diagram: o-------o A_2 gives the Coxeter group consisting of symmetries of the equilateral triangle - by which I mean all reflections and rotations.
		For example, the Dynkin diagram A_n has n dots in a row like this: o-------o-------o-------o-------o and this gives the symmetry groups of projective geometry: the geometry of points, lines, planes, and so on up to dimension n.
	math.ucr.edu /home/baez/twf_ascii/week186 (2951 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The reason this is so fun is that we can also get a group from a Dynkin diagram without choosing a field: the so-called Coxeter group!Amazingly, all sorts of formulas about this Coxeter group are special cases of formulas about simple algebraic groups over F_{q}.
		In both cases, these incidence geometries have one type of geometrical figure for each dot in the Dynkin diagram, and one basic incidence relation for each edge.
		If you give me a Dynkin diagram and a field, I will give you a simple algebraic group G. If you pick a subset of the dots in this diagram, I will give you a subgroup P of G, called a "parabolic subgroup".
	www.infomag.ru:8081 /dbase/B003E/021010-239.txt (1721 words)

	Simple Lie group - Wikipedia, the free encyclopedia
		The groups SO(p,q,R) and SO(p+q,R), for example, give rise to different real Lie algebras, but having the same Dynkin diagram.
		The Dynkin diagram has two nodes that are not connected.
		there is an 'exotic' symmetry of the diagram, corresponding to so-called triality.
	en.wikipedia.org /wiki/Simple_Lie_group (498 words)

	Table
		The column ``Roots'' gives the Dynkin diagram of the norm 2 vectors of L arranged into orbits under Aut(L).
		The group Aut(L) is a split extension R.G where R is the Weyl group of the Dynkin diagram and G is isomorphic to the subgroup of Aut(D) fixing u.
		Otherwise the letter a, d, or e is the first letter of the Dynkin diagram of the norm 0 vector, and its height is given by height(u)
	math.berkeley.edu /~reb/lattices/table2.html (350 words)

	Citations: The maximal subgroups of the classical groups - Dynkin (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		It relies on the decomposition where v 0 is a highest weight vector and L low is the sum of all weight spaces except the highest one.
		Dynkin diagrams with multiple bonds arise from folding simply laced diagrams along symmetries.
		D (G) denotes the marked Dynkin diagram, that is, the Dynkin diagram of G with the node representing the....
	citeseer.ifi.unizh.ch /context/496568/0 (1742 words)

	Citebase - Mirror symmetry and Exact Solution of 4D N=2 Gauge Theories I
		In section 8 we give a uniform treatment for quivers based on affine ADE Dynkin diagrams, by studying the mirror of elliptic ADE singularities in 2-complex dimensions (this gives the first derivation for the affine A and D cases and a third derivation for the affine E case as the quiver).
		If si denote the Dynkin numbers associated with each node of the Dynkin diagram, and if we denote the i-th 2-cycle class by Ci the 2-cycle class of the elliptic fiber E can be represented by E= s i Ci Note that this is consistent with the fact that E · E = 0.
		As discussed before an interesting case involves the configuration of the A-groups arranged along the affine ADE Dynkin diagrams, where the rank of the SU gauge group is proportional (with fixed proportionality) to the Dynkin number of the corresponding node.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:hep-th/9706110 (8855 words)

	Weyl Groups
		The polytope corresponding to the A2 Lie algebra by The McKay Correspondence is a triangle
		Note that the cuboctahedron is the Root Vector Diagram of the 12+3=15-dimensional D3 Lie algebra.
		The polytope corresponding to the E6 Lie algebra by The McKay Correspondence is not formed, as is the D6 polytope, by a set of 4 points placed on the 1-dimensional equator of a 2-dimensional sphere S2.
	www.valdostamuseum.org /hamsmith/Weyl.html (5287 words)

	PlanetMath: classification of indecomposable root systems
		The following table indicates the cardinality of and the Lie algebras and Dynkin diagrams corresponding to the above root systems.
		Cross-references: Dynkin diagrams, Lie algebras, cardinality, crystallographic, standard basis, inner product, Euclidean, perpendicular, subspace, Euclidean space, dimension, root systems, indecomposable, infinite
		This is version 2 of classification of indecomposable root systems, born on 2005-08-20, modified 2005-08-20.
	planetmath.org /encyclopedia/ClassificationOfIndecomposableRootSystems.html (136 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		One of the most striking results in the theory of cluster algebras due to S.Fomin and A.Zelevinsky is the classification of cluster algebras of finite type, which turns out to be identical to the Cartan-Killing classification.
		Fomin and Zelevinsky proved that a diagram is of finite type if and only if it is mutation-equivalent to a Dynkin diagram.
		The list contains all extended Dynkin diagrams but also has 6 more infinite series, and a substantial number of exceptional diagrams with at most 9 vertices.
	www-math.mit.edu /~combin/abstracts/apr04/seven.html (158 words)

	Why Spin(0,8)?
		----------------------------------------------- The D3 Dynkin diagram is, where v denotes the conventional linear vector representation, a denotes the adjoint representation, ad s+ and s- denote the two half-spinor representations: s+
		Note that the D3 Dynkin diagram can be written equivalently as 4 - 6 - 4 (conventional linear vector) or 4 - 4 - 4 (conformal) which shows the isomorphism between D3 and A3.
		The D4-D5-E6-E7 physics model is the natural Feynman Checkerboard theory based on the octonionic 8-real-dimensional E8 lattice with 4-complex-dimensional Witting polytope vertex figure, reduced to a 4-real-dimensional D4 lattice with 24-cell vertex figure.
	www.valdostamuseum.org /hamsmith/why8.html (3810 words)

	The 4-ality model for and (Site not responding. Last check: 2007-10-21)
		are those associated to the three simple roots corresponding to the three ends of the Dynkin diagram, and to the highest root.
		The right hand column gives a representative for each orbit, whose weighted Dynkin diagram can be found on the same line.
		Note that the Hasse diagram is almost a chain, except that there are two triples of nilpotent orbits, permuted by the triality automorphisms.
	www.maths.warwick.ac.uk /~bww/nilpotentseries/node3.html (240 words)

	P.P. Cook's Tangent Space: E11 in the Reference Frame
		The longest line of ten roots in the E11 Dynkin diagram is the A10 used, and is often called the gravity line.
		SL(11,R) is made, and then by including the eleventh generator from E11 (the one from the node that sticks out on the Dynkin diagram) this algebra is enlarged to GL(11,R).
		The decomposition gives an infinite number of generators, classified by their Dynkin labels, and in particular the Dynkin label of the eleventh root which is called the level.
	ppcook.blogspot.com /2005/05/e11-in-reference-frame.html (1577 words)

	Dynkin Digraphs (Site not responding. Last check: 2007-10-21)
		The Dynkin digraph has vertices 1,..., n; whenever c_(ij)<0 there is an edge from i to j labeled by the value -c_(ij).
		In the literature, the term Dynkin diagram is used, but we reserve this for a printed display of the Dynkin digraph (or Coxeter graph) corresponding to a finite or affine Coxeter group (see Section Finite and Affine Coxeter Groups below).
		A Dynkin digraph has an edge from i to j if, and only if, it has an edge from j to i (although the labels may be different); hence strong and weak connectivity are equivalent for these graphs.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text1012.htm (300 words)

	Citations: Semisimple subalgebras of semisimple Lie algebras - Dynkin (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Definition 2.9 The unique element ff 2 R with maximal height is called the longest root (or highest root) of R. The extended Dynkin diagram of R is formed by adding one further node to the Dynkin....
		, Ch.1, 2) proving the proposition reduces to prove it for the fundamental weights where the Dynkin numbers are known from Dynkin (6] Table 5) We reproduce his numbers for the convenience of the reader (and correct some misprints in his table) Ar d# i = r 1 i 1 for i =1, r.
		As a general reference for algebraic groups we cite Borel s book [2] and for information on root systems we refer the reader to Bourbaki [3] The simple roots in a base of a root system of a simple algebraic group are indexed in accordance with loc.
	citeseer.ist.psu.edu /context/272283/0 (2264 words)

	Ein neuer Zusammenhang zwischen einfachen Gruppen und einfachen Singularitaeten (Site not responding. Last check: 2007-10-21)
		Simple singularities are classified by simply-laced Dynkin diagrams (type ADE) which in turn define certain simple algebraic groups.
		Let G be a simple algebraic group with simply-laced Dynkin diagram.
		This singularity is simple with the same Dynkin diagram as G. Furthermore, "perturbing x" is a versal deformation of the singularity.
	www.math.rutgers.edu /~knop/papers/einf.html (206 words)

	Classification of d-Complete Posets (Site not responding. Last check: 2007-10-21)
		Let's say that a Dynkin diagram is of general type E if it is Y-shaped, has exactly one branch of length 1, and is not of type A. There are 15 classes of possible slant irreducible components.
		In all but Classes 1-3, the top tree of each slant irreducible component is of general type E. Each of the 15 classes is further divided into subclasses, and each subclass possesses a unique maximal member.
		Each poset in one of the other classes corresponds to a certain especially nice kind of Weyl group element in a simply laced Kac-Moody Weyl group whose Dynkin diagram is of general type E. The order diagrams shown on the following pages are the order diagrams for the unique maximal poset within each class.
	www.math.unc.edu /Faculty/rap/Classif.html (366 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		While this correction might be important in applications of our work, the results of the paper are not affected by it.
		For a large class of such Dynkin diagram automorphisms, we can describe various aspects of these maps in terms of another Kac-Moody algebra, the `orbit Lie algebra'.
		In particular, the generating function for the trace of the map on modules, the `twining character', is equal to a character of the orbit Lie algebra.
	www.thphys.uni-heidelberg.de /cgi-bin/abstracts/hep-th:9506135 (208 words)

	Mathematics Colloquium --** (Site not responding. Last check: 2007-10-21)
		Abstract: There is a classical bijection between finite subgroups of SL_2(C) and Dynkin diagrams of ADE type.
		The classification of finite subgroups of SL_2(C) is essentially based on the classification of symmetry groups of (possibly degenerate) regular polyhedra.
		In this talk, starting from a finite subgroup of SL_2(C), we will present and compare two parallel ways of constructing the basic representation of an affine/toroidal Lie algebra whose Dynkin diagram corresponds to the finite group in the classical manner.
	www.math.virginia.edu /~colloq/2000-01/00-12-08-wang.html (115 words)

	Niemeier lattice - Wikpedia (Site not responding. Last check: 2007-10-21)
		Niemeier lattices are usually labeled by the Dynkin diagram of their root systems.
		These Dynkin diagrams have rank either 0 or 24, and all of their components have the same Coxeter number.
		(The Coxeter number, at least in these cases, is the number of roots divided by the dimension.) There are exactly 24 Dynkin diagrams with these properties, and there turns out to be a unique Niemeier lattice for each of these Dynkin diagrams.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Niemeier_lattice (187 words)

	Simple Lie group (Site not responding. Last check: 2007-10-21)
		Firstly, for example, the SO(p,q,R) and SO(p+q,R) give rise to different Lie algebras with the same Dynkin diagram.
		1 Classification by Lie algebra and Dynkin diagram
		See also Cartan matrix, Coxeter matrix, Dynkin diagram, Weyl group, Coxeter group, Kac-Moody algebras.
	www.theezine.net /s/simple-lie-group.html (261 words)

	Citations: Unzerlegbare Darstellungen - Gabriel (ResearchIndex)
		Then S is of finite representation type if and only if the underlying graph of the quiver Delta is one of the Dynkin graphs A n (n 1) D n (n 4) E n (8 n 6) 5.2 Tame and wild representation type To define....
		and [3] that is characterized by the Dynkin quivers and dim furnishes a bijection between the isomorphism classes of the indecomposable modules and the positive roots of the corresponding simple Lie algebra g: It is natural to ask whether it is possible to recover a Lie algebra structure in.
		Under the assumption that the spectrum of A is finite, the pairs of bounded self adjoint operators A, B satisfy semi linear relation (22) if and only if the support of B is contained in the graph of this relation, constructed with respect to oe(A) We give one of the....
	citeseer.ist.psu.edu /context/136495/0 (1948 words)

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