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Topic: Cartan decomposition

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Cartan connection

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Henri Cartan

Einstein Cartan theory

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	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 (7012 words)

	Pauli matrices - Wikipedia, the free encyclopedia
		form a Cartan pair of the Lie algebra su(2).
		Now, a version of Cartan decomposition states that any element U in the Lie group SU(2) can be expressed in the form
		In that context, the Cartan decomposition given above is called the Z-Y decomposition of a single-qubit gate.
	en.wikipedia.org /wiki/Pauli_matrices (769 words)

	Lie derivative
		Note that the Cartan formulation of the "Lie derivative" acting on differential forms is precisely a cohomological method.
		IMO Cartan's methods and cohomology are important methods of making progress in areas where other techniques have failed.
		The idea that Cartan's formula for the Lie derivative acting on forms yields the functional format for the Lorentz force as a component of the Lie derivative is mathematically correct.
	quantumfuture.net /quantum_future/lie.htm (8445 words)

	APPENDIX G
		This appendix collects formulae that express certain quantities associated with the Hilbert space Hilb(n) in terms of those similarly calculated for its "positive" and "negative" subspaces.
		This division is invariantly defined under the adjoint action of the group U(G, n) by its Cartan canonical decomposition.
		Consider influence on equation (13.8b) by the restriction to a subspace, enforced in the n, k> basis.
	graham.main.nc.us /~bhammel/FCCR/apdxG.html (345 words)

	Huajun's Homepage (Site not responding. Last check: 2007-10-02)
		Given a representation of a real or complex reductive Lie group G on V, a parabolic subgroup P imposes certain partial order (or grading) on V. One can describe the parabolic subgroup orbits and invariants on V by studying the structure of G in accordance with the partial order.
		They are relative to the Bruhat decomposition, Iwasawa decomposition, and some topics in representation theory.
		Among them are Jordan decomposition, complete multiplicative Jordan decomposition, Cartan decomposition, Bruhat decomposition, Iwasawa decomposition, and Levi decomposition.
	www.auburn.edu /~huanghu (233 words)

	Amazon.com: Complex Semisimple Lie Algebras: Books: Jean-Pierre Serre,G.A. Jones (Site not responding. Last check: 2007-10-02)
		The conjugacy of Cartan subalgebras, which enables us to define the numeric invariant called rank, is developed in analogous way to the book of Chevalley [Thorie des Groupes de Lie, 1951].
		Bases of roots and their elementary properties are developed, and how to go from a basis to another by emans of the Weyl group [it is supposed that the root system is reduced, for nonreduced systems see for example the sixth chapter of Bourbaki: Algbres de Lie, Hermann 1967].
		Chapter six begin with the classical Weyl theorems, and the Cartan decomposition of a semisimple Lie algebra is obtained.
	www.amazon.com /exec/obidos/tg/detail/-/3540678271?v=glance (1212 words)

	Qubiter
		The CS decomposition is very closely related to the Generalized Singular Value Decomposition (GSVD).
		To perform the CS decomposition, Qubiter calls subroutines from Lapack, a free software package available at Netlib.
		"A constructive algorithm for the Cartan decomposition of SU(2^N)", by H. Sa Earp, J. Pachos, quant-ph/0505128
	www.ar-tiste.com /qubiter.html (1032 words)

	Home page of Domenico D'Alessandro (Site not responding. Last check: 2007-10-02)
		For two level quantum systems a decomposition due to F. Lowenthal is used.
		This decomposition is a generalization of the classical Euler decomposition for SU(2).
		For the four level quantum system of two interacting spin 1/2 particles an algorithm is also presented which uses a modification of Cartan decomposition of the Lie group SU(4) and allows to drive of the state to any desired final configuration.
	www.public.iastate.edu /~daless/QC5.html (135 words)

	Abstract of P. Gerardin's talk (Site not responding. Last check: 2007-10-02)
		A rank l Riemannian symmetric space Z is a homogeneous space under the group G of real points of a connected reductive group.
		The points in Z correspond to the Cartan involutions on G, the maximal flat subspaces in Z correspond to the maximal split tori for G, the distinguished boundary of Z to the set of minimal parabolic subgroups.
		The Cartan decomposition realizes a map from G\(ZxZ) to a real cone of dimension l, and this defines spheres on the space Z. The ring of invariant differential operators on Z is a rank l polynomial algebra.
	logic.pdmi.ras.ru /GeneralSeminar/abstr/056.html (201 words)

	[No title]
		Via the $p$-adic Bruhat decomposition for $G_{K}$, the computation of $\Phi(g)$ is reduced to that of $\Phi(\xi(\pi))$ where $\xi \in \mbox{Hom}({\bf G_{m}},T)$ is as in \cite{I2} or \cite{M1}.
		We are able to parametrize the simple generalized Igusa local zeta functions associated to $(G^{'},\rho)$ by examining the above Cartan product decomposition of $\rho$.
		The generalized Igusa local zeta function $Z_{K}(s)$ associated to $(G^{'},\rho)$ is simple if and only if the decomposition diagram of $(G^{'},\rho)$ is connected.
	www.maths.tcd.ie /EMIS/journals/ERA-AMS/1995-03-003/1995-03-003.tex.html (926 words)

	Publications of Jorn B. Olsson (Site not responding. Last check: 2007-10-02)
		On Cartan matrices and lower defect groups for covering groups of symmetric groups (with
		A note on Cartan matrices for symmetric groups (with
		Decomposition matrices for spin characters of symmetric groups at characteristic 3 (with
	www.math.ku.dk /~olsson/links/publ.htm (611 words)

	Research
		For each group G, a maximal compact subgroup K is found.
		The Lie algebra k, of K is a Cartan subalgebra of g and determines a Cartan decomposition of g, which in turn, gives a decomposition of g into the root subspaces with respect to k.
		Then by considering the Laplacian and Casimir operators in polar coordinates, the decomposition of L
	euphrates.wpunj.edu /faculty/llarullm/research.htm (627 words)

	PlanetMath:
		ANCOVA (in general linear model) owned by CWoo
		An example for Schur decomposition owned by georgiosl
		anticommutator bracket (in Cartan Calculus) owned by plinko
	planetmath.org /encyclopedia/A (2167 words)

	Amazon.ca: Complex Semisimple Lie Algebras: Books (Site not responding. Last check: 2007-10-02)
		The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras.
		The theory is illustrated by using the example of sln; in particular, the representation theory of sl2 is completely worked out.
		This concerns solvable and nilpotent Lie algebras, as well as some generic results on semisimple algebras (results that do not involve Cartan subalgebras).
	www.amazon.ca /exec/obidos/ASIN/3540678271 (982 words)

	TAMU Math Preprints
		Siegel's lemma with additional conditions, submitted January 2005.
		On effective Witt decomposition and Cartan-Dieudonné theorem, submitted January 2005.
		Kornelson, K. Dykema, D. Freeman, D. Larson, M. Ordower, and E. Weber, Ellipsoidal tight frames and projection decomposition of operators, submitted.
	www.math.tamu.edu /news_events/newsletter/0pubs.html (1613 words)

	Papers for research purposes (Site not responding. Last check: 2007-10-02)
		Key words: Iwasawa decomposition, Lie algebra, Radon transform, inverse of the Abel (or Radon) transform, symmetric space of non-compact type, noncompact, Cartan decomposition, spherical Fourier transform, Abel transform, Helgason, Lie groups, root system.
		Key words: Iwasawa decomposition, Lie algebra, SL(n, $R$), inverse of the Abel (or Radon) tLaplacian (Beltrami-Laplace operator), Helgason, Cartan decomposition, Harish-Chandra, non-hypergeometric functions.
		Key words: noncompact, symmetric space, Iwasawa decomposition, Abel transform, Lie algebra, Radon transform, Helgason, semisimple Lie groups (or algebras), Cartan decomposition, SL(n, $R$).
	www.cs.laurentian.ca /sawyer/list.html (5256 words)

	HMC Math 196: Lie Groups and Lie Algebras, Fall 2001---Assignments (Site not responding. Last check: 2007-10-02)
		The Cartan Decomposition, the Killing Form, and the Weyl Group
		Prove that the set of all n by n matrices (a
		Prove or disprove: If g is a simple non-trivial Lie algebra over the complex numbers and h is a Cartan subalgebra of g, then [hh] = 0.
	www.math.hmc.edu /~orrison/teaching/m196f01/assignments.html (83 words)

	seminar_abstracts (Site not responding. Last check: 2007-10-02)
		We first give a classification of equivariant compactifications of symmetric varieties.
		The method itself might be of interest: we use an analogue of Cartan decomposition for the p-adic orbit space G(k[[t]])\(G/G^\sigma)(k((t))).
		We identify a nice class of compactifications called by various names: maximal Satake compactification, Oshima-Sekiguchi compactification and De Concini-Procesi compactification.
	www.wam.umd.edu /~jda/seminar_abstracts.html (388 words)

	Table of contents for Library of Congress control number 2005050811 (Site not responding. Last check: 2007-10-02)
		Bibliographic record and links to related information available from the Library of Congress catalog
		Cartan decomposition and the Generalized elastic problems 19 Chapter 3.
		Cartan algebras, root spaces and extra integrals of motion 92 Chapter 9.
	www.loc.gov /catdir/toc/fy0604/2005050811.html (128 words)

	Neeb: Globality in semisimple Lie groups
		In Section 2 we develop some algebraic tools concerning real root decompositions with respect to compactly embedded Cartan algebras and invariant cones in semisimple Lie algebras.
		In Section 3 these tools, combined with the results from Section 1, yield a characterization of those invariant cones in a semisimple Lie algebra
		[M] Compactifications of symmetric spaces, II, The Cartan domains, Amer.
	www.numdam.org /numdam-bin/item?id=AIF_1990__40_3_493_0 (392 words)

	[No title] (Site not responding. Last check: 2007-10-02)
		Lenny Fukshansky: "Effective Structure Theorems for Quadratic Spaces"
		Two of the best known structure theorems for quadratic spaces over global fields are Witt decomposition and Cartan-Dieudonne' theorems.
		I will also talk about the effective versions of these theorems that I recently proved.
	www.math.tamu.edu /~cherylr/postdocs.html (854 words)

	Time Travel Portal :: View topic - Warp-Drive Quantum Computation
		Posted: Sat Dec 04, 2004 1:54 pm Subject: 3862
		Recently it has been shown that time-optimal quantum computation is attained by using the Cartan decomposition of a unitary matrix.
		We extend this approach by noting that the unitary group is compact.
	timetravelportal.com /viewtopic.php?t=1290 (254 words)

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