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# Topic: Lie group decompositions

###### In the News (Tue 21 May 19)

 Lie group decompositions - Wikipedia, the free encyclopedia They are essential technical tools in the representation theory of Lie groups and Lie algebras; they can also be used to study the algebraic topology of such groups and associated homogeneous spaces. Since the use of Lie group methods became one of the standard techniques in twentieth century mathematics, many phenomena can now be referred back to decompositions. The same ideas are often applied to Lie groups, Lie algebras, algebraic groups and p-adic number analogues, making it harder to summarise the facts into a unified theory. en.wikipedia.org /wiki/Lie_group_decompositions   (276 words)

 Infinite-dimensional Lie groups and their representations   (Site not responding. Last check: 2007-09-16) The investigation of the infinite-dimensional Lie groups is partially motivated by the role they play in connection with the description of the symmetries of certain structures arising in the mathemathical physics. According to the general principle of Lie theory, the Lie groups are studied by means of the corresponding Lie algebras. These are the so-called enlargible Lie algebras, and their name comes from the fact that any'' Lie algebra gives rise to a local Lie group, and the difficult point is to enlarge'' the local group to a group in the usual sense. www.imar.ro /~dbeltita/int1.html   (311 words)

 Talk Abstract: Conserving algorithms on Lie groups   (Site not responding. Last check: 2007-09-16) Lie groups and their tangent or cotangent bundles form a class of manifolds for which particularly natural generalizations of these procedures exist. The usual weighted averages of vector field evaluations are computed in the Lie algebra or its dual using the natural trivializations of the tangent and cotangent bundles; the resulting update vector is then mapped into the phase space using either the exponential map or an approximate `algorithmic' exponential map. Group decompositions can be used to design efficient algorithmic exponentials; these decompositions are often suggested by the symmetries of the relevant mechanical system. www.ima.umn.edu /dynsys/wkshp_abstracts/lewis1.html   (268 words)

 lie group In mathematics, a Lie group (pronounced "lee", named after Sophus Lie) is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. Lie groups are important in mathematical analysis, physics and geometry because they serve to describe the symmetry of analytical structures. If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite dimensional Lie algebra g over F there is a unique (up to isomorphism) simply connected Lie group G with g as Lie algebra. www.fact-library.com /lie_group.html   (1285 words)

 Lie group decompositions   (Site not responding. Last check: 2007-09-16) Since the use of Lie group methods became one of the standard techniques in twentieth century mathematics, many phenomena can now be referred back todecompositions. The same ideas are often applied to Lie groups, Lie algebras, algebraic groups and p-adic number analogues,making it harder to summarise the facts into a unified theory. The Iwasawa decomposition KAN of a semisimple group generalises the way a square real matrix can be written as aproduct of an orthogonal matrix and an upper triangular matrix (a consequence of orthogonalization). www.therfcc.org /lie-group-decompositions-329322.html   (248 words)

 Partitioned Runge-Kutta methods in Lie-group setting (ResearchIndex)   (Site not responding. Last check: 2007-09-16) Next, we equip the (co)tangent bundle of a Lie group with a group structure and treat it as a Lie group. The structure of the differential equations on the (co)tangent-bundle Lie group is such that partitioned versions of the... Recurrence relation for the factors in the polar decomposition on.. citeseer.ist.psu.edu /331146.html   (372 words)

 Description of LiE   (Site not responding. Last check: 2007-09-16) LiE is the name of a software package that enables mathematicians and physicists to perform computations of a Lie group theoretic nature. It focuses on the representation theory of complex semisimple (reductive) Lie groups and algebras, and on the structure of their Weyl groups and root systems. LiE does not compute directly with elements of the Lie groups and algebras themselves; it rather computes with weights, roots, characters and similar objects. young.sp2mi.univ-poitiers.fr /~marc/LiE/description.html   (237 words)

 Harish-Chandra Info - Bored Net - Boredom   (Site not responding. Last check: 2007-09-16) During this period he established as his special area the study of the discrete series representations of semisimple Lie groups - which are the closest analogue of the Peter-Weyl theory in the non-compact case. The methods were formidable and inductive, using Lie group decompositions. He is also known for work with Armand Borel founding the theory of arithmetic groups; and for papers on finite group analogues www.borednet.com /e/n/encyclopedia/h/ha/harish_chandra.html   (148 words)

 Lie Theoretic Comments   (Site not responding. Last check: 2007-09-16) In 1993 we noticed that the simplest cases of the Lakshmibai-Seshadri basis results, those for Demazure modules whose lowest weights were integer multiples of weights obtained by acting on lambda with a lambda-minuscule w, could be given purely combinatorial interpretations and proofs. The ways of reading off the elements of the poset which are consistent with the partial order on the poset correspond to the reduced decompositions of w. Hence the number of reduced decompositions of w is equal to the number of order extensions of P(w). www.math.unc.edu /Faculty/rap/Lie.html   (333 words)

 Rutgers Lie Group Seminar Fall 1998 Friday, Oct. 9, 1998, 11:30am-12:30pm, Hill 425 Alan Weinstein: Near-homomorphisms from compact Lie groups A near-homomorphism from a group G to a group H is a map a from G to H such that a(gh) is "sufficiently close" to a(g)a(h) for all g and h in G. When G and H are compact Lie groups, it was shown by Grove and Karcher in the 1970's that a near-homomorphism is near a homomorphism. The affine flag manifolds of G are the spaces of parahoric subalgebras (a local field analogue of parabolic subalgebras) of a given type in the Lie algebra with scalars extended to F. www.math.rutgers.edu /~knop/seminar/Seminar_Fall98.html   (1210 words)

 [ref] 61 Lie Algebras A Lie algebra L is an algebra such that xx=0 and x(yz)+y(zx)+z(xy)=0 for all x,y,z ∈ L. is the (Lie) derived subalgebra of the Lie algebra Representations of Lie algebras are dealt with in the same way as representations of ordinary algebras (see Representations of Algebras). www.gap-system.org /Manuals/doc/htm/ref/CHAP061.htm   (4974 words)

 [ref] 60 Lie Algebras In this section we describe two functions that calculate a direct sum decomposition of a Lie algebra; the so-called Levi decomposition and the decomposition into a direct sum of ideals. is the quotient of the free Lie algebra Representatios of Lie algebras are delat with in the same way as representations of ordinary algebras (see Representations of Algebras). www.mathematik.uni-kassel.de /gap4/ref/CHAP060.htm   (4428 words)

 List of Lie group topics - Wikipedia, the free encyclopedia See Table of Lie groups for a list The special unitary group SU(1,1) is the unit sphere in the ring of coquaternions.It is the group of hyperbolic motions of the Poincare disk model of the hyperbolic plane. See also: List of harmonic analysis and representation theory topics en.wikipedia.org /wiki/List_of_Lie_group_topics   (94 words)

 Lie Group Beyond: An Introduction, 2nd Edition   (Site not responding. Last check: 2007-09-16) Lie Groups Beyond an Introduction takes the reader from the end of introductory Lie group theory to the threshold of infinite-dimensional group representations. A feature of the presentation is that it encourages the reader's comprehension of Lie group theory to evolve from beginner to expert: initial insights make use of actual matrices, while later insights come from such structural features as properties of root systems, or relationships among subgroups, or patterns among different subgroups. Representations of sl(2,C) Elementary Theory of Lie Groups www.booksmatter.com /b0817642595.htm   (342 words)

 Algorithms for Quantum Control Based on Decompositions of Lie Groups - D'Alessandro (ResearchIndex)   (Site not responding. Last check: 2007-09-16) Abstract: Algorithms for the control of quantum mechanical systems are presented which are based on decompositions of Lie groups. ...the values in the closed one dimensional Lie group generated by A. In conclusion wehave T c 2 p. Constructivecontrollability for quantum mechanical systems based on decompositions of Lie groups such as the one in (14) is explored in citeseer.ist.psu.edu /421163.html   (396 words)

 Lie Group Structures on Quotient Groups and Universal Complexifications for Infinite Dimensional Lie Groups ...   (Site not responding. Last check: 2007-09-16) Lie Group Structures on Quotient Groups and Universal Complexifications for Infinite Dimensional Lie Groups (ResearchIndex) Lie Group Structures on Quotient Groups and Universal Complexifications for Infinite Dimensional Lie Groups (2001) 38 Lie Groups and Lie Algebras (context) - Bourbaki - 1989 citeseer.ist.psu.edu /451751.html   (485 words)

 DECISION, CONTROL, AND OPTIMIZATION SEMINAR   (Site not responding. Last check: 2007-09-16) The describing models are bilinear systems whose state varies on the Lie group of special unitary matrices of dimensions two and four, respectively. By performing decompositions of Lie groups, taking into account the describing equations at hand, we derive control laws to steer the state of the system to any desired final configuration. The analysis is motivated by the implementation of one and two quantum bit logic gates using nuclear magnetic resonance experiments which involve the active manipulation of one or two interacting spin $\frac{1}{2}$ particles. black.csl.uiuc.edu /seminars/dco_seminar/PastSeminars/01SpringSeminars/dalessandro.html   (134 words)

 CWI Tract   (Site not responding. Last check: 2007-09-16) On one hand, better ways are brought forward to satisfy the physicists' demands to collect explicit data about representations, tensor product decompositions etc., while, on the other hand, new impulses are given to effective computations of invariants of groups acting on given spaces and even invariants of elements pertaining to these groups. Omar Foda and Jan Sanders exploited the use of Lie group notions and techniques in computations for statistical mechanics and differential equations, respectively. The remaining two contributions are somewhat further away from the theory of Lie groups. www.cwi.nl /publications/Abstracts_tracts/tr-84.html   (244 words)

 Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations   (Site not responding. Last check: 2007-09-16) Using Lie theory, we generalize this to an n-qubit decomposition, the concurrence canonical decomposition (CCD) SU(2{sup n})=KAK. The group K fixes a bilinear form related to the concurrence, and in particular any unitary in K preserves the tangle vertical bar <{phi} vertical bar (-i{sigma}{sub 1}{sup y}){center_dot}{center_dot}{center_dot}(-i{sigma}{sub n}{sup y}) vertical bar {phi}> vertical bar{sup 2} for n even. Thus, the CCD shows that any n-qubit unitary is a composition of a unitary operator preserving this n-tangle, a unitary operator in A which applies relative phases to a set of GHZ states, and a second unitary operator which preserves the tangle. www.osti.gov /energycitations/product.biblio.jsp?osti_id=20547162   (403 words)

 Complexity theory for Lie-group solvers - Celledoni, Iserles, Nrsett, Orel (ResearchIndex)   (Site not responding. Last check: 2007-09-16) Abstract: Commencing with a brief survey of Lie-group theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Lie-group structure: Runge--Kutta--Munthe-Kaas schemes, Fer and Magnus expansions. 25 Approximating the exponential from a Lie algebra to a Lie gr.. 12 Lectures on Lie Groups and Lie Algebras (context) - Carter, Segal et al. citeseer.lcs.mit.edu /celledoni99complexity.html   (883 words)

 Harish Chandra   (Site not responding. Last check: 2007-09-16) He then moved to the USA, where he was at ColumbiaUniversity from 1950 to 1963. During this period he established as his special area the study of the discrete seriesrepresentations of semisimple Lie groups - which are theclosest analogue of the Peter-Weyl theory in the non-compactcase. He is also known for work with Armand Borel founding the theory of arithmetic groups ; and for papers on finite group analogues www.therfcc.org /harish-chandra-164547.html   (153 words)

 Sonia G Schirmer's Research Dissipative Groups and the Bloch Ball A. Solomon and S. Schirmer, in Group 24: Physical and Mathematical Aspects of Symmetries (ISBN 0-75-030933-4) 2003 Group theoretical aspects of control of quantum systems, International Symposium on Symmetries in Science (Bregenz, Austria, Aug 2001) Kinematical bounds on the optimization of observables, the question of their dynamical realizability and Lie groups XXIII Colloquium on Group Theoretical Methods in Physics (Dubna, Russia, Aug 2000) cam.qubit.org /users/sonia/research/research.html   (1523 words)

 Generalized Polar Decompositions for the Approximation of the Matrix Exponential Generalized Polar Decompositions for the Approximation of the Matrix Exponential: SIAM Journal on Matrix Analysis and Applications Vol. The algorithms presented have the property that, if $Z \in {\frak{g}}$, a Lie algebra of matrices, then the approximation for exp(Z) resides in G, the matrix Lie group of ${\frak{g}}$. We propose algorithms based on a splitting of Z into matrices having a very simple structure, usually one row and one column (or a few rows and a few columns), whose exponential is computed very cheaply to machine accuracy. epubs.siam.org /sam-bin/dbq/article/37755   (249 words)

 Algorithms reading group homepage   (Site not responding. Last check: 2007-09-16) Satish will first talk about a theorem regarding the decomposition of planar graphs that appeared in a paper by Klein, Plotkin and Rao (get it here). This result follows from the Okamura-Seymour theorem on multicommodity flows in planar graphs where all the terminals lie on the same face, which shows that max-flow equals min-cut in such case. In fact, the statement of the theorem is much stronger, i.e., it gives the existance of disjoint paths. cs.berkeley.edu /~chrishtr/research/archive/spring2002/algorithms.html   (800 words)

 [No title] V.G. Kac, "Infinite dimensional Lie algebras" (Cambridge 1990, 3rd ed.). Derive the character table for the possible matrix representations of this group. and identify the Dynkin diagram and hence the Lie algebras obtained. fy.chalmers.se /~tfebn/LieAlgprogrammeMaster2004.html   (885 words)

 lie Smaller highways continue on from the end of the LIE to Greenport on the North Fork and past the Hamptons to Montauk on the South Fork. Cynics have suggested that the acronym is appropriate, in that the term "expressway" is a lie. In 1999, an HOV lane was added from Deer Park to (near) Hicksville. www.fact-library.com /lie.html   (135 words)

 Mathematica (Tm) Packages for Computing Principal Decompositions of Simple Lie Algebras and Applications in Extended ... The main goal of the first package, SimpleLieAlgebras.m, is to determine the principal three-dimensional subalgebra of a simple Lie algebra. The package provides several functions which give some elements related to simple Lie algebras (generators in fundamental and adjoint representation, roots, Killing form, Cartan matrix, etc.). The second package, PrincipalDecomposition.m, concerns the principal decomposition of a Lie algebra with respect to the principal three-dimensional embedding. library.wolfram.com /infocenter/Articles/5286   (162 words)

 Efficient Computation of the Matrix Exponential by Generalized Polar Decompositions In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie-group structure. Our point of departure is the method of generalized polar decompositions, which we modify and combine with similarity transformations that bring the underlying matrix to a form more amenable to efficient computation. We develop techniques valid for a range of Lie groups: the orthogonal group, the symplectic group, Lorentz, isotropy, and scaling groups. epubs.siam.org /sam-bin/dbq/article/41593   (190 words)

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