Semi-simple real subalgebras of non-compact semi-simple real Lie algebras IV

The investigation of the problem of embedding a semi-simple real Lie algebra $\text{L}$′ in a non-compact semi-simple real Lie algebra $\text{L}$ is extended to the case in which at least one of the real Lie algebras has a semi-simple complex extension, which consists of the direct sum of two simple complex Lie algebras. Detailed procedures are given, which together with those given previously, allow the construction of all embeddings of $\text{L}$′ in $\text{L}$ when their complex extensions are A₁, B₁, C₁, D₁ or a direct sum of any two of these. The procedures are illustrated by considering examples corresponding to complex Lie algebra embeddings A₁?(A₂⊕A₂), (A₁⊕A₁)?(A₂⊕A₂), (A₁⊕A₁)?A₃, (A₁⊕A₁)??(A₃⊕A₃) and (A₁⊕A₁)?(A₃⊕A₂). Because of its physical significanc embeddings of SL(2,C) in simple and semi-simple real Lie algebras are studied in detail.     

Step algebras of semi-simple subalgebras of Lie algebras

For each pair (G,K) where G is a complex finite-dimensional Lie algebra and K a semi-simple subalgebra of G, we construct an associative algebra (step algebra) $\text{Y}$ (G,K) and a homomorphism i_*: $\text{Y}$ (G,K)→E(G) is the enveloping algebra of G. $\text{Y}$ (G,K) has the following properties: (1) If V is any G-module and x ? V a K-maximal vector, then sx = i_* (s)x is K-maximal for any s ? $\text{Y}$ (G,K); (2) If V is irreducible and a certain simple criteria is fulfilled, then any K-maximal vector can be written in the form sx_m, s ? $\text{Y}$ (G,K), where x_m is some fixed K-maximal vector. Because of these properties $\text{Y}$ (G,K) has great practical value when constructing irreducible representations of Lie algebras in a form which makes the reduction with respect to a semi-simple subalgebra explicit.     

Positive cones in enveloping algebras

The concept of strong ordering on enveloping algebras of finite-dimensional Lie algebras is introduced and studied as a generalization of the corresponding notion for the commutative polynomial algebra. A linear functional f on an enveloping algebra $\text{E}$ (G) is called strongly positive if f(x) ? 0 for all x ? $\text{E}$(G) which are mapped on positive operators for all G-integrable irreducible representations of $\text{E}$(G). We prove that for each real connected Lie group G ≠ R₁ there are positive, not strongly positive, linear functionals on $\text{E}$(G). A non-commutative problem of moments is defined. It has a solution iff the corresponding linear functional is strongly positive.     

Extensions of representations and cohomology

In this article we study the extensions of Banach space representations of a Lie group G. We introduce different spaces of 1-cohomology on G, or on its Lie algebra $\text{G}$, and make the connection between these spaces and the equivalence (or weak equivalence) classes of extensions.We characterize, from the properties of the 1-cohomology groups, the spaces of differentiable and analytic vectors of an extension and prove a kind of Whitehead's lemma.For Lie groups with a large compact subgroup K, we specialize to K-finite representations, and introduce and study Naimark equivalence of extensions.The results are applied to classify the extensions of the irreducible representations of $\text{G =}\text{SL}\text{(2,}\text{R}\text{)}$.     

Analytical structure and properties of Coulomb wave functions for real and complex energies

The radical Coulomb wave functions are analysed in their dependence on the energy E considered as a complex parameter. Repulsive and attractive fields are both considered. First turning to the function Φ_l ∝ r^?l?1F_l introduced by Briet, slightly modifying its definition, and assuming that the angular momentum is also a complex parameter, for which the notation L is used, it is proved that Φ_L is an entire function of both E and L. From an expansion of the regular Whittaker function given by Buchholz, the Taylor expansion of Φ_L in powers of E and a simple recurrence relation for its coefficients are easily obtained. The expansion of the regular function F_l is readily obtained from that of Φ_L for L = l, but the irregular function G_l contains Φ_l and $\text{?Φ}{}_{\mathrm{L}}\text{?L}$ for L = l and ?l?1. Having proved that the expansion obtained for Φ_L in powers of E can also be regarded as a uniformly convergent series of entire functions of L, the derivative $\text{?Φ}{}_{\mathrm{L}}\text{?L}$ can be obtained by term-by-term derivation. This method for obtaining the expansion of G_l is straightforward and leads to a final result involving essentially: (i) the conventional function $\text{h(η) =}\text{1}\text{2}\text{ψ(1 + iη) +}\text{1}\text{2}\text{ψ(1 ? iη) ?}\text{ln}\text{η}$ which is singular at η = ∞, i.e., at k = 0; (ii) two entire functions of E, namely Φ_l and Ψ_l; the terms of the expansion of the latter in powers of E contain only Bessel functions multiplied by Bernoulli numbers and coefficients easily obtained from a simple recurrence relation. As an application of the above results, the last sections contain: (i) an alternate from of G_l expansion useful in numerical computations; (ii) the definition and expansion of two linearly independent solutions of the Coulomb equation which are entire in E; (iii) the expansion and threshold properties of the outgoing and incoming solutions, O_l and I_l, corresponding to those we have obtained for F_l and G_l.     

Affine extensions of supersymmetry: The finite case

We examine the graded Poincaré (GP) Lie algebra of supersymmetry with a view to constructing possible affine extensions of the algebra, i.e. extensions of the GP algebra which contain as a subalgebra the Lie algebra $\text{ga}\text{(4,}\text{R}\text{)}$. We restrict our attention in this paper to an examination of the finite extensions. We demonstrate explicitly that if we adjoin only a symmetric tensor generator to the GP algebra, then such a generator cannot generate all the deformations, in particular the shear, of the general affine group GA(4, $\text{R}$). Similarly, we show that adjoining the supersymmetry generator to $\text{ga}\text{(4,}\text{R}\text{)}$ cannot lead to closure of the resulting algebra, even in the trivial case. We further demonstrate that the GLA $\text{ga}\text{(}\text{4}\text{4}\text{,}\text{R}\text{)}$ does not contain the Lie algebra $\text{ga}\text{(4,}\text{R}\text{)}$ represented over the entire superspace upon which $\text{ga}\text{(}\text{4}\text{4}\text{,}\text{R}\text{)}$ is defined.     

States on the current algebra

We introduce the field algebra Σ$\text{D}$(M;ⁿ?ⁿ$\text{g}$) associated with the current algebra $\text{D}$_r(M;$\text{g}$) for the Lie algebra $\text{g}$ over physical space M. The Heisenberg magnet model is generalized to this continuum. It is shown that the Hamiltonian can be given meaning as implementing a derivation of the field algebra in certain representations.We introduce new representations of the current algebra. For example, if G = SU(2), a representation in L²(R³)?³ is [σ(?)F]^j = ε^jkl?_kψ_l for (?_k) = ? in $\text{D}$_r(M;$\text{g}$)(ψ_l = F. This has cyclic subrepresentations with prime parts.     

Invariant star-products on symplectic manifolds

Let (M,F) be a symplectic manifold and consider a Lie subalgebra $\text{G}$ of its Lie algebra of symplectic vector fields. We prove that every one-differentiable deformation of order k of the Poisson Lie algebra of M, which is invariant with respect to $\text{G}$, extends to an invariant one-differentiable deformation of infinite order. If M admits a $\text{G}$-invariant linear connection, a similar result holds true for differentiable deformations and for star-products. In particular, if M admits a $\text{G}$- -invariant linear connection, there always exists a $\text{G}$-invariant star-product.     

Matter fields and metric deformations in multidimensional unified theories

This paper describes a first study of the effects due to including matter fields in generalized Kaluza-Klein (KK) theories with nonabelian compact gauge group G and nontrivial fibres $\text{V}$^K. The approach is based on the first-order Einstein-Cartan (EC) general relativity in (4 + K) dimensions. In the EC theory there are two basic mechanisms which can lead to a spontaneously compactified KK background geometry R⁴ × $\text{V}$^K: (A) a particular kind of energy-momentum density matter condensate in the quantized ground state, or (B) a particular kind of spin-density matter condensate. If (A) or (B) are operating, the inconsistencies usually found between the KK ansatz and the matter-free EC theory are avoided. Mechanism (B) works only when $\text{V}$^K is parallelizable. It is shown that the expansion of matter fields in normal modes on $\text{V}$^K implies that one must include deformations of the Yang-Mills (YM) potentials contained in the usual KK metrics. We discuss and characterize one class of such deformations. As a case study, we consider fibres $\text{V}$^K ～ G′, where G′ is a semisimple compact Lie group. We allow for the “maximal” YM gauge group G_L′ × G_R′. We carry out the harmonic analysis for spinor fields and study the mass spectrum and YM quantum numbers of the normal modes. We rely on mechanism (B) to provide a curvature-free connection (“parallelization”) on $\text{V}$^KG′ by means of a suitable vertical constant torsion. Minimal YM couplings are of size $\text{l}\text{L}\text{≡ g}$, where l is the Planck length and L is the length of the fibre; nonminimal YM couplings are of size L. Nonzero masses are of size L^?1. The massless modes are found and discussed. There would be no massless modes if the parallelizing vertical torsion were absent. This torsion also implies the vanishing of the cosmological constant. When the theory is restricted to massless modes, the YM deformations disappear and the dimensional reduction to four dimensions yields an effective YM theory, which is renormalizable at energies far below L^?1: the effective theory is obtained by letting L → 0 with g ? 1 fixed and by neglecting all masses of order L^?1; g corresponds to the bare YM coupling constant. The surviving effective YM gauge group is G_L′ and the matter fields are in a particular representation of G_L′ × G_R′, corresponding to the zero mass eigenvalue. Explicit examples are discussed for G′ = SU(2) and G′ = SU(3).     

The moment map and collective motion

Given a Hamiltonian action of a Lie group G on a symplectic manifold M there is an induced map Φ: $\text{M → g}{}^1$ where $\text{g}{}^1$ is the dual space to the Lie algebra, g, of G. The map Φ is called the moment map. Any function P on $\text{g}{}^1$ then gives rise to a function F = P ° Φ on M which is a “collective Hamiltonian” associated to the group action G. We show how the rigid rotor, liquid drop, and other collective models of the nucleus fit into this framework. We describe the steps involved in integrating collective equations of motion and indicate some principles involved in the choice of collective Hamiltonians, i.e., the functions P. We discuss these constructions in some detail for the case that G is a semidirect product.     

An effective method of investigation of positive maps on the set of positive definite operators

Let $\text{A}$₁ be the algebra of linear operators on the n-dimensional Hilbert space $\text{H}$₁, and let $\text{A}$₂ be the algebra of linear operators of the m-dimensional Hilbert space $\text{H}$₂. Let $\text{L}$($\text{A}$₁, $\text{A}$₂) denote the complex space of linear maps from $\text{A}$₁ to $\text{A}$₂. By a positive map we mean an element of the space $\text{L}$($\text{A}$₁, $\text{A}$₂) (superoperator with respect to $\text{H}$₁) which maps positive definite operators in $\text{A}$₁ into positive definite operators in $\text{A}$₂. The aim of this paper is to present an effective method which allows to verify whether a given superoperator Λ∈$\text{L}$($\text{A}$₁, $\text{A}$₂) is a positive map. Besides that necessary and sufficient conditions for the positive definiteness of even-degree forms in many variables are given.     

Symmetric spaces of exceptional groups

We address the problem of the reasons for the existence of 12 symmetric spaces with the exceptional Lie groups. The 1 + 2 cases for G₂ and F₄, respectively, are easily explained from the octonionic nature of these groups. The 4 + 3 + 2 cases on the E_6,7,8 series require the magic square of Freudenthal and, for the split case, an appeal to the supergravity chain in 5, 4, and 3 space—time dimensions.     

Variational calculations of realistic models of nuclear matter

We report variational calculations of nuclear matter with a semi-realistic Reid v₁₂ model, and a realistic v₁₄ model of the two-nucleon interaction operator. The v₁₄ model fits the available nucleon-nucleon scattering data up to 425 MeV lab energy, and has relatively weak L² and $\text{(}\text{L}\text{·}\text{S}\text{)}{}^2$ interactions in addition to the standard central, tensor and ($\text{L}\text{·}\text{S}$). The L² and $\text{(}\text{L}\text{·}\text{S}\text{)}{}^2$ interactions are treated semiperturbatively; their contribution reduces the overbinding of nuclear matter. However, the equilibrium k_F = 1.7 fm^?1 and E₀ = ?17.5 MeV obtained with the v₁₄ model are both higher than their empirical values k_F = 1.33 fm^? and E₀ = ?16 MeV. We assume that the difference between the calculated and empirical E(ρ) is entirely due to three-nucleon interactions (TNI). The TNI contributions are phenomenologically added to the nuclear matter energy, and their parameters are adjusted to obtain the correct equilibrium energy, density and compressibility. The required TNI contributions appear to be of reasonable magnitude.     

Classification of Finite Irreducible Modules over the Lie Conformal Superalgebra CK 6

We classify all continuous degenerate irreducible modules over the exceptional linearly compact Lie superalgebra E(1, 6), and all finite degenerate irreducible modules over the exceptional Lie conformal superalgebra CK ₆, for which E(1, 6) is the annihilation superalgebra.     

Extension of the hidden symmetry algebra in classical principal chiral models

By considering the inverse Brézin-Itzykson-Zinn-Justin-Zuber procedure and the linearization system derived from it, the infinite-parameter Lie algebra for the hidden symmetry in 2-dimensional principal chiral and superchiral models is extended from $\text{G}\text{?}\text{C}\text{[t]}\text{to G}\text{?}\text{C}\text{[t,t}{}^{\mathrm{?1}}\text{] ⊕}\text{G}$, where $\text{G}\text{?}\text{C}\text{[t,t}{}^{\mathrm{?1}}\text{]}$ is the true full Kac-Moody algebra without center. This is done in the formalism with left-invariant potentials. In the parallel development with right-invariant potentials it is shown to be impossible to enlarge the hidden symmetry algebra further.     

The application of tailored modulation techniques to depth profiling with auger electron spectroscopy

Tailored modulation techniques (TMT) are applied to depth profiling to eliminate errors in signal strength measurements caused by Auger line shape changes. The application of TMT is illustrated by profiling through Al₂ O₃Al interfaces. When peak-to-peak heights in first derivative spectra $\text{n}$⁽¹⁾_m (E) are used for profiling, the measured Al KL_2,in3L_2,3 signal strength shows a large decrease near the interface. This artifact is reduced when peak heights in $\text{n}$_m (E) are used but can be eliminated only when Auger area values are used for profiling. The peak heights in $\text{n}$_m (E) and Auger area values can both be obtained in real time with TMT and plotted automatically with conventional multiplexing equipment. Some features of Auger spectra obtained using TMT are illustrated, and a comparison of depth profiles obtained using peak-to-peak heights in $\text{n}$¹_m (E), peak heights in $\text{n}$_m (E) and Auger area values demonstrates the usefulness of TMT in depth profiling.     

Study of Ag/Si(111) submonolayer interface: I. Electronic structure by angle-resolved UPS

Angle-resolved ultraviolet photoelectron spectra have been measured for well defined Ag/Si(111) submonolayer interfaces of (1) $\text{Si(111)(}\text{3}\text{×}\text{3}\text{)R30°-Ag}$, (2) “Si(111)(6 × 1)-Ag”, and (3) Ag/Si(111) as deposited at room temperature. Non-dispersive and very narrow (FWHM ～ 0.4–0.5 eV) Ag 4d derived peaks are found at 5.6 and 6.5 eV below the Fermi level for surface (1) and at 5.3 and 6.0 eV for surface (2). Dispersions of sp “binding” states in the energy range between E_F and Ag 4d states have been precisely determined for surface (1). Electronic structures similar to those of the Ag(111) surface, including the surface state near E_F, have been observed for surface (3).