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Mapping the geometry of the E6 group

Description: In this paper we present a construction for the compact form of the exceptional Lie group E{sub 6} by exponentiating the corresponding Lie algebra e{sub 6}, which we realize as the sum of f{sub 4}, the derivations of the exceptional Jordan algebra J{sub 3} of dimension 3 with octonionic entries, and the right multiplication by the elements of J{sub 3} with vanishing trace. Our parameterization is a generalization of the Euler angles for SU(2) and it is based on the fibration of E{sub 6} via a F{sub 4} subgroup as the fiber. It makes use of a similar construction we have performed in a previous article for F{sub 4}. An interesting first application of these results lies in the fact that we are able to determine an explicit expression for the Haar invariant measure on the E{sub 6} group manifold.
Date: October 1, 2007
Creator: Cerchiai , Bianca; Bernardoni, Fabio; Cacciatori, Sergio L.; Cerchiai, Bianca L. & Scotti, Antonio
Partner: UNT Libraries Government Documents Department

Sixth-Order Lie Group Integrators

Description: In this paper we present the coefficients of several 6th order symplectic integrator of the type developed by R. Ruth. To get these results we fully exploit the connection with Lie groups. This integrator, as well as all the explicit integrators of Ruth, may be used in any equation where some sort of Lie bracket is preserved. In fact, if the Lie operator governing the equation of motion is separable into two solvable parts, the Ruth integrators can be used.
Date: March 1, 1990
Creator: Forest, E.
Partner: UNT Libraries Government Documents Department

Invariance, groups, and non-uniqueness: The discrete case

Description: Lie group methods provide a valuable tool for examininginvariance and non-uniqueness associated with geophysical inverseproblems. The techniques are particularly well suited for the study ofnon-linear inverse problems. Using the infinitesimal generators of thegroup it is possible to move within the null space in an iterativefashion. The key computational step in determining the symmetry groupsassociated with an inverse problem is the singular value decomposition(SVD) of a sparse matrix. I apply the methodology to the eikonal equationand examine the possible solutions associated with a crosswelltomographic experiment. Results from a synthetic test indicate that it ispossible to vary the velocity model significantly and still fit thereference arrival times. the approach is also applied to data fromcorosswell surveys conducted before and after a CO2 injection at the LostHills field in California. The results highlight the fact that a faultcross-cutting the region between the wells may act as a conduit for theflow of water and CO2.
Date: March 24, 2005
Creator: Vasco, D.W.
Partner: UNT Libraries Government Documents Department

Mapping the geometry of the F4 group

Description: In this paper, we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3 x 3 Hermitian matrices with octonionic entries. We use a parametrization which generalizes the Euler angles for SU(2) and is based on the fibration of F4 via a Spin(9) subgroup as a fiber. This technique allows us to determine an explicit expression for the Haar invariant measure on the F4 group manifold. Apart from shedding light on the structure of F4 and its coset manifold OP2 = F4/Spin(9), the octonionic projective plane, these results are a prerequisite for the study of E6, of which F4 is a (maximal) subgroup.
Date: May 28, 2007
Creator: Bernardoni, Fabio; Cacciatori, Sergio L; Scotti, Antonio & Cerchiai, Bianca L.
Partner: UNT Libraries Government Documents Department

Exceptional groups, symmetric spaces and applications

Description: In this article we provide a detailed description of a technique to obtain a simple parameterization for different exceptional Lie groups, such as G{sub 2}, F{sub 4} and E{sub 6}, based on their fibration structure. For the compact case, we construct a realization which is a generalization of the Euler angles for SU(2), while for the non compact version of G{sub 2(2)}/SO(4) we compute the Iwasawa decomposition. This allows us to obtain not only an explicit expression for the Haar measure on the group manifold, but also for the cosets G{sub 2}/SO(4), G{sub 2}/SU(3), F{sub 4}/Spin(9), E{sub 6}/F{sub 4} and G{sub 2(2)}/SO(4) that we used to find the concrete realization of the general element of the group. Moreover, as a by-product, in the simplest case of G{sub 2}/SO(4), we have been able to compute an Einstein metric and the vielbein. The relevance of these results in physics is discussed.
Date: March 31, 2009
Creator: Cerchiai, Bianca L. & Cacciatori, Sergio L.
Partner: UNT Libraries Government Documents Department

Anarchy and hierarchy

Description: We advocate a new approach to study models of fermion massesand mixings, namely anarchy proposed in hep-ph/9911341. In this approach,we scan the O(1) coefficients randomly. We argue that this is the correctapproach when the fundamental theory is sufficiently complicated.Assuming there is no physical distinction among three generations ofneutrinos, the probability distributions in MNS mixing angles can bepredicted independent of the choice of the measure. This is because themixing angles are distributed according to the Haar measure of the Liegroups whose elements diagonalize the mass matrices. The near-maximalmixings, as observed in the atmospheric neutrino data and as required inthe LMA solution to the solar neutrino problem, are highly probable. Asmall hierarchy between the Delta m2 for the atmospheric and the solarneutrinos is obtained very easily; the complex seesaw case gives ahierarchy of a factor of 20 as the most probable one, even though thisconclusion is more measure-dependent. U_e3 has to be just below thecurrent limit from the CHOOZ experiment. The CP-violating parameter sindelta is preferred to be maximal. We present a simple SU(5)-likeextension of anarchy to the charged-lepton and quark sectors which workswell phenomenologically.
Date: September 14, 2000
Creator: Haba, Naoyuki & Murayama, Hitoshi
Partner: UNT Libraries Government Documents Department

Minimality of the Special Linear Groups

Description: Let F denote the field of real numbers, complex numbers, or a finite algebraic extension of the p-adic field. We prove that the special linear group SLn(F) with the usual topology induced by F is a minimal topological group. This is accomplished by first proving the minimality of the upper triangular group in SLn(F). The proof for the upper triangular group uses an induction argument on a chain of upper triangular subgroups and relies on general results for locally compact topological groups, quotient groups, and subgroups. Minimality of SLn(F) is concluded by appealing to the associated Lie group decomposition as the product of a compact group and an upper triangular group. We also prove the universal minimality of homeomorphism groups of one dimensional manifolds, and we give a new simple proof of the universal minimality of S∞.
Date: December 1997
Creator: Hayes, Diana Margaret
Partner: UNT Libraries

A Topological Uniqueness Result for the Special Linear Groups

Description: The goal of this paper is to establish the dependency of the topology of a simple Lie group, specifically any of the special linear groups, on its underlying group structure. The intimate relationship between a Lie group's topology and its algebraic structure dictates some necessary topological properties, such as second countability. However, the extent to which a Lie group's topology is an "algebraic phenomenon" is, to date, still not known.
Date: August 1997
Creator: Opalecky, Robert Vincent
Partner: UNT Libraries

Euler angles for G2

Description: We provide a simple parameterization for the group G2, which is analogous to the Euler parameterization for SU(2). We show how to obtain the general element of the group in a form emphasizing the structure of the fibration of G2 with fiber SO(4) and base H, the variety of quaternionic subalgebras of octonions. In particular this allows us to obtain a simple expression for the Haar measure on G2. Moreover, as a by-product it yields a concrete realization and an Einstein metric for H.
Date: March 10, 2005
Creator: Cacciatori, Sergio; Cerchiai, Bianca Letizia; della Vedova,Alberto; Ortenzi, Giovanni & Scotti, Antonio
Partner: UNT Libraries Government Documents Department

Duality, Entropy and ADM Mass in Supergravity

Description: We consider the Bekenstein-Hawking entropy-area formula in four dimensional extended ungauged supergravity and its electric-magnetic duality property. Symmetries of both"large" and"small" extremal black holes are considered, as well as the ADM mass formula for N=4 and N=8 supergravity, preserving different fraction of supersymmetry. The interplay between BPS conditions and duality properties is an important aspect of this investigation.
Date: February 23, 2009
Creator: Cerchiai, Bianca L.; Ferrara, Sergio; Marrani, Alessio & Zumino, Bruno
Partner: UNT Libraries Government Documents Department

Dimensions, induces and congruence classes of representations of affine Kac-Moody algebras (with examples for affine E/sub 8/)

Description: Affine generalizations of some familiar notions from the representation theory of semisimple Lie algebras/groups are introduced, described and illustrated. The multiplicity of a weight and the dimension congruence class, and indices of a representation are touched upon. Examples of the highest weight representations of affine E/sub 8/ are considered as a preview of far more extensive results of this type to appear.
Date: January 1, 1986
Creator: Kass, S.N. & Patera, J.
Partner: UNT Libraries Government Documents Department

Twining characters and orbit Lie algebras

Description: We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.
Date: December 5, 1996
Creator: Fuchs, Jurgen; Ray, Urmie; Schellekens, Bert & Schweigert, Christoph
Partner: UNT Libraries Government Documents Department

Notes on lie algebraic analysis of achromats

Description: Normal form technique is a powerful method to analyze the achromat problem. Assume the one cell map M{sub cell} = ARe{sup :h{sub 3}}:{sub e}{sup :h{sub 4}}: A{sup {minus}1}, where h{sub 3},h{sub 4} are the normal forms of the generators of the unit cell map, and A is the nonlinear transformation that brings M{sub cell} into its normal form; then the map of the whole system is M{sub N} = M{sub cell}{sup N} = AR{sup N} A{sup {minus}1} = I, provided that we can set e{sup :h{sub 3}}:, e{sup :h{sub 4}}, and R{sup N} to the identity (or only {delta} dependent) maps. Therefore, the conditions to form an achromat are h{sub 3} and h{sub 4} equal to zero (or {delta} dependent only) and the total linear map is identity. In this report, we will apply these conditions to a FODO array (a simple model system) to make it an achromat. We will start from Hamiltonians and work all the way up to obtain the analytical expressions of the required sextupole and octupole strengths.
Date: January 1, 1995
Creator: Wang, Chunxi & Chao, A.
Partner: UNT Libraries Government Documents Department

A formula for the integration of radiation using Yoshida's Lie methods

Description: We present our earlier formula and show how the nonsymplectic operator for the classical radiation can be imported in a generalized Yoshida's formula. The challenge is to break the force into exactly solvable parts. Then the entire machinery of Lie integrators takes over.
Date: October 1, 1998
Creator: Forest, E., Parsa, Z.
Partner: UNT Libraries Government Documents Department

Nonlinear instability and chaos in plasma wave-wave interactions. II. Numerical methods and results

Description: In Part I of this work and Physics of Plasmas, June 1995, the behavior of linearly stable, integrable systems of waves in a simple plasma model was described using a Hamiltonian formulation. It was shown that explosive instability arises from nonlinear coupling between modes of positive and negative energy, with well-defined threshold amplitudes depending on the physical parameters. In this concluding paper, the nonintegrable case is treated numerically. Several sets of waves are considered, comprising systems of two and three degrees of freedom. The time evolution is modelled with an explicit symplectic integration algorithm derived using Lie algebraic methods. When initial wave amplitudes are large enough to support two-wave decay interactions, strongly chaotic motion destroys the separatrix bounding the stable region for explosive triplets. Phase space orbits then experience diffusive growth to amplitudes that are sufficient for explosive instability, thus effectively reducing the threshold amplitude. For initial amplitudes too small to drive decay instability, small perturbations might still grow to arbitrary size via Arnold diffusion. Numerical experiments do not show diffusion in this case, although the actual diffusion rate is probably underestimated due to the simplicity of the model.
Date: May 1, 1995
Creator: Kueny, C.S. & Morrison, P.J.
Partner: UNT Libraries Government Documents Department

Solutions of the Noh Problem for Various Equations of State Using Lie Groups

Description: A method for developing invariant equations of state for which solutions of the Noh problem will exist is developed. The ideal gas equation of state is shown to be a special case of the general method. Explicit solutions of the Noh problem in planar, cylindrical and spherical geometry are determined for a Mie-Gruneisen and the stiff gas equation of state.
Date: August 1, 1998
Creator: Axford, R. A.
Partner: UNT Libraries Government Documents Department

Quantum groups, braiding matrices and coset models

Description: We discuss a few results on quantum groups in the context of rational conformal field theory with underlying affine Lie algebras. A vertex-height correspondence - a well-known procedure in solvable lattice models - is introduced in the WZW theory. This leads to a new definition of chiral vertex operator in which the zero mode is given by the q-Clebsch Gordan coefficients. Braiding matrices of coset models are found to factorize into those of the WZW theories. We briefly discuss the construction of the generators of the universal enveloping algebra in Toda field theories. 21 refs., 2 figs.
Date: July 1, 1989
Creator: Itoyama, H.
Partner: UNT Libraries Government Documents Department

Physical interpretation of supercoherent states and their associated Grassmann numbers

Description: A physical interpretation of supercoherent states is suggested. It is based upon the observation that an ordinary coherent state is an eigenstate of a specific mode of the radiation field. A supercoherent state is viewed as a photino coherently combined with photons of the same mode. An interpretation of the associated Grassmann-valued numbers of the state is also discussed. 13 refs.
Date: January 1, 1991
Creator: Nieto, M.M.
Partner: UNT Libraries Government Documents Department

Soft-edged magnet models for higher-order beam-optics map codes

Description: Continuously varying surface and volume source-density distributions are used to model magnetic fields inside of cylindrical volumes. From these distributions, a package of subroutines computes on-axis generalized gradients and their derivatives at arbitrary points on the magnet axis for input to the numerical map-generating subroutines of the Lie-algebraic map code Marylie. In the present version of the package, the magnet menu includes: 1. cylindrical current-sheet or radially thick current distributions with either open boundaries or with a surrounding cylindrical boundary with normal field lines (which models high-permeability iron), 2. Halbach-type permanent mutipole magnets, either as sheet magnets or as radially thick magnets, 3. modeling of arbitrary fields inside a cylinder by use of a fictitious current sheet. The subroutines provide on-axis gradients and their z derivatives to essentially arbitrary order, although in the present 3rd and 5th order Marylie only the 0th through 6th derivatives are needed. The formalism is especially useful in beam-optics applications, such as magnetic lenses, where realistic treatment of fringefield effects is needed.
Date: January 1, 2002
Creator: Walstrom, P. L. (Peter L.)
Partner: UNT Libraries Government Documents Department

Analytic second- and third-order achromat designs

Description: An achromat is a transport system that carries a beam without distorting its transverse phase space distribution. In this study, we apply the Lie algebraic technique to a repetitive FODO array to make it either a second-order or a third-order achromat. (Achromats based on reflection symmetries are not studied here.) We consider third-order achromats whose unit FODO cell layout is shown. The second-order achromat layout is the same, except the octupoles are absent. For the second-order achromats, correction terms (due to the finite bending of the dipoles) to the well-known formulae for the sextupole strengths are derived. For the third-order achromats, analytic expressions for the five octupole strengths are given. The quadrupole, sextupole and octupole magnets are assumed to be thin-lens elements. The dipoles are assumed to be sector magnets filling the drift spaces. More details of the analysis have been reported elsewhere.
Date: July 1, 1995
Creator: Wang, Chunxi & Chao, Alex
Partner: UNT Libraries Government Documents Department

Quantum groups: Geometry and applications

Description: The main theme of this thesis is a study of the geometry of quantum groups and quantum spaces, with the hope that they will be useful for the construction of quantum field theory with quantum group symmetry. The main tool used is the Faddeev-Reshetikhin-Takhtajan description of quantum groups. A few content-rich examples of quantum complex spaces with quantum group symmetry are treated in details. In chapter 1, the author reviews some of the basic concepts and notions for Hopf algebras and other background materials. In chapter 2, he studies the vector fields of quantum groups. A compact realization of these vector fields as pseudodifferential operators acting on the linear quantum spaces is given. In chapter 3, he describes the quantum sphere as a complex quantum manifold by means of a quantum stereographic projection. A covariant calculus is introduced. An interesting property of this calculus is the existence of a one-form realization of the exterior differential operator. The concept of a braided comodule is introduced and a braided algebra of quantum spheres is constructed. In chapter 4, the author considers the more general higher dimensional quantum complex projective spaces and the quantum Grassman manifolds. Differential calculus, integration and braiding can be introduced as in the one dimensional case. Finally, in chapter 5, he studies the framework of quantum principal bundle and construct the q-deformed Dirac monopole as a quantum principal bundle with a quantum sphere as the base and a U(1) with non-commutative calculus as the fiber. The first Chern class can be introduced and integrated to give the monopole charge.
Date: May 13, 1996
Creator: Chu, C. S.
Partner: UNT Libraries Government Documents Department

Lie group applications to the solution of differential equations

Description: This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). The project applied Lie group techniques to the solution of differential equations (DEs) describing physical systems important to LANL`s scientific mission. There were two main objectives: (1) to construct analytic or quasianalytic solutions for use as benchmark test problems; and (2) to develop improved numerical solution algorithms that possess the same symmetries (e.g. rotational, translational, and scaling invariance) as the DEs. Significant progress was achieved on both these objectives.
Date: August 1, 1997
Creator: Cranfill, C.W.; Coggeshall, S.V.; Knapp, C.E.; Smitherman, D.P.; Caramana, E.J. & Axford, R.A.
Partner: UNT Libraries Government Documents Department

Invariants and labels for Lie-Poisson Systems

Description: Reduction is a process that uses symmetry to lower the order of a Hamiltonian system. The new variables in the reduced picture are often not canonical: there are no clear variables representing positions and momenta, and the Poisson bracket obtained is not of the canonical type. Specifically, we give two examples that give rise to brackets of the noncanonical Lie-Poisson form: the rigid body and the two-dimensional ideal fluid. From these simple cases, we then use the semidirect product extension of algebras to describe more complex physical systems. The Casimir invariants in these systems are examined, and some are shown to be linked to the recovery of information about the configuration of the system. We discuss a case in which the extension is not a semidirect product, namely compressible reduced MHD, and find for this case that the Casimir invariants lend partial information about the configuration of the system.
Date: April 1, 1998
Creator: Thiffeault, J.L. & Morrison, P.J.
Partner: UNT Libraries Government Documents Department

Relabeling symmetries in hydrodynamics and magnetohydrodynamics

Description: Lagrangian symmetries and concomitant generalized Bianchi identities associated with the relabeling of fluid elements are found for hydrodynamics and magnetohydrodynamics (MHD). In hydrodynamics relabeling results in Ertel`s theorem of conservation of potential vorticity, while in MHD it yields the conservation of cross helicity. The symmetries of the reduction from Lagrangian (material) to Eulerian variables are used to construct the Casimir invariants of the Hamiltonian formalism.
Date: April 1, 1996
Creator: Padhye, N. & Morrison, P.J.
Partner: UNT Libraries Government Documents Department