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Topic: Cartan matrix

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	Matrix in TutorGig Encyclopedia
		The Gnosis is a hovercraft in the Matrix trilogy.
		The Novalis is a hovercraft in the Matrix trilogy.
		The Mjolnir is a hovercraft in the Matrix trilogy.
	www.tutorgig.com /es/Matrix/1 (977 words)

	Élie Cartan - Wikipedia, the free encyclopedia
		Cartan was born in Dolomieu in Savoie, and became a student at the École Normale Superieure in Paris in 1888.
		Cartan writes of the influence on him of Riquier’s general PDE theory.
		This is constantly seen in areas such as calculus of variations, Bäcklund transformations and the general theory of differential systems; roughly speaking those parts of differential algebra which feel that the existing, Galois theory-led model of symmetry is too narrow and requires something more analogous to a category of relations.
	en.wikipedia.org /wiki/Elie_Cartan (687 words)

	List of matrices (Site not responding. Last check: 2007-10-09)
		Companion matrix - the companion matrix of a polynomial is a special form of matrix, whose eigenvalues are equal to the roots of the polynomial.
		Permutation matrix - matrix representation of a permutation.
		Toeplitz matrix - a matrix with constant diagonals.
	www.worldhistory.com /wiki/L/List-of-matrices.htm (825 words)

	Special unitary group - Wikipedia, the free encyclopedia
		The special unitary group is a subgroup of the unitary group U(n), consisting of all n×n unitary matrices, which is itself a subgroup of the general linear group GL(n, C).
		The weight eigenvectors are the Cartan subalgebra itself and the matrices with only one nonzero entry which is off diagonal.
		Even though the Cartan subalgebra h is only n − 1 dimensional, to simplify calculations, it is often convenient to introduce an auxiliary element, the unit matrix which commutes with everything else (which should not be thought of as an element of the Lie algebra!) for the purpose of computing weights and that only.
	en.wikipedia.org /wiki/SU(3) (686 words)

	CSDC : Frame vs. Metric connections, and their curvatures
		Then by exterior differential and algebraic processes a right Cartan matrix of connection 1-forms is produced, and a second application of exterior differential and algbraic processes to the Cartan connection produces the concept of a Cartan matrix of curvature 2-forms.
		The Cartan connection is independent from the choice of metric on the final state, the Christoffel connection depends upon the metric.
		When the Jacobian matrix is used to play the role of a Frame matrix, then the Right Cartan Connection coefficients defined on the initial state, computed not from the metric but from the Frame field, are exactly equal to the Christoffel Connection coefficients deduced from the pullback metric on the initial state.
	www22.pair.com /csdc/ed3/ed3fre26.htm (2632 words)

	Élie Cartan -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a (The Romance language spoken in France and in countries colonized by France) French (A person skilled in mathematics) mathematician, who did fundamental work in the theory of (additional info and facts about Lie group) Lie groups and their geometric applications.
		Cartan added the (additional info and facts about exterior derivative) exterior derivative, as an entirely geometric and coordinate-independent operation.
		With these basics – Lie groups and differential forms – he went on to produce a very large body of work, and also some general techniques such as (additional info and facts about moving frame) moving frames, that were gradually incorporated into the mathematical mainstream.
	www.absoluteastronomy.com /encyclopedia/_/l/%e9lie_cartan.htm (1037 words)

	Weyl Groups
		A Cartan matrix is: 2 0 -1 0 2 -1 -1 -1 2 The Coxeter-Dynkin diagram is
		A Cartan matrix is: 2 0 -1 0 0 0 2 -1 0 0 -1 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 The Coxeter-Dynkin diagram is
		A Cartan matrix is: 2 0 -1 0 0 0 0 2 0 -1 0 0 -1 0 2 -1 0 0 0 -1 -1 2 -1 0 0 0 0 -1 2 -1 0 0 0 0 -1 2 The Coxeter-Dynkin diagram is
	www.valdostamuseum.org /hamsmith/Weyl.html (5287 words)

	APPENDIX B
		A Cartan subalgebra of a semisimple Lie algebra L given the canonical decomposition, is a maximal Abelian subalgebra of M with which the symmetric space G/K is homeomorphic.
		The Cartan subalgebra of u(n) is a Lie algebra of diagonal Hermitean matrices, that of su(n) is a Lie algebra of Hermitean matrices of trace zero.
		The Cartan metric metric provides a map between L and its dual space L^+, the space of linear functionals acting on L. The adjoint action of the group on L is then mapped by the Cartan metric to an action of the group on L^+, the coadjoint representation.
	graham.main.nc.us /~bhammel/FCCR/apdxB.html (7008 words)

	PlanetMath: Cartan matrix
		The Cartan matrix uniquely determines the root system, and is unique up to simultaneous permutation of the rows and columns.
		It is also the basis change matrix from the basis of fundamental weights to the basis of simple roots in
		This is version 1 of Cartan matrix, born on 2002-12-20.
	www.planetmath.org /encyclopedia/CartanMatrix.html (103 words)

	Weyl Groups
		A Cartan matrix, which determines the commutation relations of the Lie algebra, is determined by the D2 root vector space is determined by ratios of the inner products of the positive roots.
		A Cartan matrix is: 2 0 0 2 Since 4-1 = 3 is the dimension of the imaginary quaternions, Spin(0,4) has quaternionic structure and in fact generates two copies of the Lie group S3, one for the group of Lorentz rotations and another for the group of Lorentz boosts.
		The commutation relations between the Cartan subalgebra element at the origin and each of the 2 elements on each side are determined by the upper and lower triangular entries of the Cartan matrix.
	www.valdostamuseum.org /hamsmith/WeyLie.html (3488 words)

	RMK articles : Shipov, Torsion, and Absolute Parallelism
		A4 spaces are 4 dimensional spaces for which both the matrix of Cartan Curvature 2-forms and the vector of Cartan Torsion 2-forms generated by exterior differentiation of the Basis Frame are globally zero.
		Note that the concept of Cartan Torsion 2-forms is not the same as the concept of torsion due to an asymmetric "affine" connection.
		The Cartan matrix [C] is equivalent to the matrix constructed from the Christoffel symbols of the induced metric on the space of parameters.
	www22.pair.com /csdc/pd2/pd2fre51.htm (755 words)

	Cartan matrices (Site not responding. Last check: 2007-10-09)
		In the noncrystallographic case, the Cartan matrix is defined over a cyclotomic extension of the rational field (Chapter CYCLOTOMIC FIELDS).
		The Cartan type of a root datum is entirely determined by its Cartan matrix.
		Returns the name of the Cartan type of the Cartan matrix C. If C is not a Cartan matrix, this gives an error message listing the rows/columns where the problem was detected.
	www.math.niu.edu /help/math/magmahelp/text460.html (408 words)

	Finite and Affine Coxeter Groups
		The Cartan name of a Coxeter matrix M, Coxeter graph G, Cartan matrix C, or Dynkin digraph D. If the corresponding Coxeter group is not finite or affine, an error is flagged.
		Print the Dynkin diagram of a Coxeter matrix M, Coxeter graph G, Cartan matrix C, Dynkin digraph D or Cartan name N. If the corresponding group is not affine or is not crystallographic, an error is flagged.
		Print the Coxeter diagram of a Coxeter matrix M, Coxeter graph G, Cartan matrix C, Dynkin digraph D or Cartan name N. If the corresponding group is not affine or is not crystallographic, an error is flagged.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text1013.htm (1093 words)

	PlanetMath: generalized Cartan matrix
		Such a matrix is called symmetrizable if there is a diagonal matrix
		Cross-references: symmetric, diagonal matrix, integers, off-diagonal entries, matrix
		This is version 2 of generalized Cartan matrix, born on 2003-08-20, modified 2003-08-21.
	planetmath.org /encyclopedia/Symmetrizable.html (68 words)

	Generalized Kac-Moody algebra (Site not responding. Last check: 2007-10-09)
		These differ from the relations of a (symmetrizable) Kac-Moody algebra mainly by allowing the diagonal entries of the Cartan matrix to be non-positive.
		A generalized Kac-Moody algebra is obtained from a universal one by changing the Cartan matrix, by the operations of killing something in the center, or taking a central extension, or adding outer derivations.
		Algebras with Lorentzian Cartan subalgebra whose denominator function is an automorphic form of singular weight.
	toshare.dynup.net /en/Generalized_Kac-Moody_Lie_algebra.htm (748 words)

	Lie groups (Site not responding. Last check: 2007-10-09)
		It is important to note that here, three different bases for vectors are used: the Cartan basis of simple roots, the Dynkin basis of fundamental weights, and some further "outer space" in which the algebra's roots live (this may be higher-dimensional, or be identical to one of the others).
		This is the number of rows of the Cartan matrix, and not actually its rank.
		Map the Cartan matrix to the symmetrized Cartan matrix (which is the metric for root space with fundamental roots as basis).
	www.cip.physik.uni-muenchen.de /~tf/lambdatensor/LambdaTensor1.1.4/doc/cartan-dynkin.html (1811 words)

	Cartan Matrices
		Then the group generated by s_1,..., s_n is a Coxeter group with Coxeter matrix M. In other words, a Cartan matrix specifies a representation of the Coxeter group as a real reflection group.
		A Cartan matrix corresponding to the Coxeter matrix M or Coxeter graph G. Note that the Cartan matrix of a Coxeter system is not unique.
		If the matrix is crystallographic however, it is defined over the integers regardless of the value of this flag.
	www.math.wayne.edu /answers/magma2.10/htmlhelp/text1011.htm (690 words)

	Construction of Coxeter Groups (Site not responding. Last check: 2007-10-09)
		A list of pairs describing the irreducible Cartan types corresponding to the matrix M. The special type texttt{X} is returned for a component which is not a valid irreducible Cartan matrix corresponding to a finite reflection group.
		It acts on the root alpha to produce alpha Cphi^t, where C is the Cartan matrix.
		Converts a permutation p representing an element of the Coxeter group W to a matrix in the reflection representation.
	www.dtr.isy.liu.se /Magma/text289.html (1707 words)

	Help Texts for coxeter and weyl, v2.4
		If R is a Coxeter matrix, cartan_matrix(R) returns the Cartan matrix for some crystallographic root system with Coxeter matrix R, if one exists.
		If R is a matrix of integers whose (1,1)-entry is 2, it is assumed that R is itself a Cartan matrix and cartan_matrix(R) returns R. For a description of root system data structures, see coxeter[structure].
		Equivalently, it is the determinant of the Cartan matrix.
	www.math.lsa.umich.edu /~jrs/software/coxeterhelp.html (8531 words)

	Introduction (Site not responding. Last check: 2007-10-09)
		The simple reflections and hence the group W are completely determined by the Cartan matrix; hence it is a natural starting point for the construction of a Coxeter group.
		Moreover the well-known classification of finite Coxeter groups means that the Cartan matrix has a very restricted form in which the irreducible blocks are completely specified by giving a emph{type} and a emph{rank}.
		A Cartan matrix (and the corresponding root system) is said to be emph{irreducible} if there is only one connected component.
	www.dtr.isy.liu.se /Magma/text288.html (815 words)

	Constructing Real Reflection Groups
		They are respectively the matrix whose rows are the simple roots, the matrix whose rows are the simple coroots, and the Cartan matrix.
		C=AB^t must be a Cartan matrix for W. It is not necessary to specify all three matrices: any two of them will determine the third.
		If C is not determined, it is taken to be the standard matrix described in Section Cartan Matrices.
	wwwmaths.anu.edu.au /research.programs/aat/htmlhelp/text1086.htm (516 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		An interesting correlation between the order of initial commutation relations and the Cartan matrix of the resulting algebra is observed.
		The second pair $k_1',k_2'$ of Cartan generators is to be identified with $k_1,k_2$.
		Second equations in (\ref{15}), as well as (\ref{16})-(\ref{18}), are associated (by means of the first equations in (\ref{15})) with $q$-deformed Serre relations produced by the Cartan matrix (\ref{19}).
	thsun1.jinr.ru /~alvladim/pap/czjp96.txt (1142 words)

	Killing (Site not responding. Last check: 2007-10-09)
		The main tools in the classification of the semisimple Lie algebras are Cartan subalgebras and the Cartan matrix both first introduced by Killing.
		It was Cartan, in his doctoral thesis submitted in 1894, who found concrete representations of all the exceptional simple Lie algebras (although he didn't work out all the details in his thesis).
		In many ways Cartan was so successful in presenting Killing's classification of the semisimple Lie algebras in rigorous and complete single work, that Killing has not received as much acclaim for his remarkable achievements as one might have expected.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Killing.html (2260 words)

	Kac-Moody algebras (Site not responding. Last check: 2007-10-09)
		One way of classifying the finite-dimensional simple Lie algebras is in terms of their Cartan matrices.
		If you slightly relax the defining conditions on the Cartan matrix, you get a wider class of algebras called the Kac-Moody algebras.
		A Cartan matrix is an nxn matrix A=(a_{ij}) with integer entries such that
	it.stlawu.edu /~dmelvill/17b/KMintro.html (411 words)

	GAP Manual: 63. Weyl Groups and Hecke Algebras
		This is a square matrix corresponding to a (finite) root system R in some Euclidean space V with (i,j)-entry given by 2(r_i,r_j)/(r_i,r_i) where r_i and r_j are fundamental roots in R.
		This Cartan matrix conversely determines the root system and hence also the corresponding Weyl group W, i.e., the finite subgroup of the full orthogonal group of V generated by the reflections along the vectors in R.
		This is a left cell consisting of three elements with the corresponding matrix of mu_(x,y), the coefficient of v^((l(y)-l(x)-1)) in the Kazhdan-Lusztig polynomial P_(x,y)(v^2).
	www.math.uiuc.edu /Software/GAP-Manual/Weyl_Groups_and_Hecke_Algebras.html (1544 words)

	WILHELM KILLING (Site not responding. Last check: 2007-10-09)
		Er promovierte 1872 bei Weierstraß über die Anwendung der Elementarteiler einer Matrix auf Oberflächen.
		Er führte die Cartan-Subalgebra, die Cartan-Matrix und die Idee des Wurzelsystems ein.
		Auf Killing geht auch die Bezeichnung charakteristische Gleichung einer Matrix zurück.
	www.toonorama.com /encyclopedia/W/Wilhelm_Killing (131 words)

	MT5827 (Site not responding. Last check: 2007-10-09)
		To be able to calculate in finite dimensional Lie algebras L over the complex numbers, work with representations of L and L-modules, find the matrix of an adjoint map associated with an element of L, find the Killing form of L.
		Be able to work with star vectors and roots, and compute the Cartan matrix of an algebra given by a Dynkin diagram.
		Calculate the angle form of a connected graph and determine whether it is positive definite.
	www-maths.mcs.st-andrews.ac.uk /ug/hon5/MT5827.shtml (211 words)

	Dynkin Digraphs (Site not responding. Last check: 2007-10-09)
		Note that functions are not given for computing the Dynkin digraph of a Coxeter matrix or Coxeter graph, since a particular choice of crystallographic Cartan matrix is required.
		The Dynkin digraph of the crystallographic Cartan matrix C. CoxeterGroupOrder(D) : GrphDir -> RngIntElt
		Z^n/Gamma where Gamma is the lattice generated by the rows of the corresponding Cartan matrix.
	www.math.lsu.edu /magma/text981.htm (309 words)

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