Quantum symmetries and the Weyl–Wigner product of group representations

In the usual formulation of quantum mechanics, groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex Hilbert space, in accordance with Wigner's theorem. In the phase-space formulation, they have real, true unitary representations in the space of square-integrable functions on phase space. Each such phase-space representation is a Weyl–Wigner product of the corresponding Hilbert space representation with its contragredient, and these can be recovered by 'factorizing' the Weyl–Wigner product. However, not every real, unitary representation on phase space corresponds to a group of automorphisms, so not every such representation is in the form of a Weyl–Wigner product and can be factorized. The conditions under which this is possible are examined. Examples are presented.

The structure of quantum Lie algebras for the classical series, B-l, C-l and D-l

The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at q = 1. We explain the relationship between the structure constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for adjoint x adjoint --> adjoint We present a practical method for the determination of these quantum Clebsch-Gordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum Lie algebras associated to the classical Lie algebras B-l, C-l and D-l. In the quantum case the structure constants of the Cartan subalgebra are non-zero and we observe that they are determined in terms of the simple quantum roots. We introduce an invariant Killing form on the quantum Lie algebras and find that it takes values which are simple q-deformations of the classical ones.

Exactly solvable quantum spin ladders associated with the orthogonal and symplectic Lie algebras

We extend the results of spin ladder models associated with the Lie algebras su(2(n)) to the case of the orthogonal and symplectic algebras o(2(n)), sp(2(n)) where n is the number of legs for the system. Two classes of models are found whose symmetry, either orthogonal or symplectic, has an explicit n dependence. Integrability of these models is shown for an arbitrary coupling of XX-type rung interactions and applied magnetic field term.

Quantum Lie algebras, their existence, uniqueness and q-antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules L-h(g) of the quantized enveloping algebras U-h(g). On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra g an abstract quantum Lie algebra g(h) independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras L-h(g) are isomorphic to an abstract quantum Lie algebra g(h). In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras L-h(g) associated to the same g are isomorphic, 2) the quantum Lie product of any Ch(B) is q-antisymmetric. We also describe a construction of L-h(g) which establishes their existence.

Integrability and exact solution for coupled BCS systems associated with the su(4) Lie algebra

We introduce an integrable model for two coupled BCS systems through a solution of the Yang-Baxter equation associated with the Lie algebra su(4). By employing the algebraic Bethe ansatz, we determine the exact solution for the energy spectrum. An asymptotic analysis is conducted to determine the leading terms in the ground state energy, the gap and some one point correlation functions at zero temperature. (C) 2002 Published by Elsevier Science B.V.

Self-interacting scalar field cosmologies: unified exact solutions and symmetries

We investigate a mechanism that generates exact solutions of scalar field cosmologies in a unified way. The procedure investigated here permits to recover almost all known solutions, and allows one to derive new solutions as well. In particular, we derive and discuss one novel solution defined in terms of the Lambert function. The solutions are organised in a classification which depends on the choice of a generating function which we have denoted by x(phi) that reflects the underlying thermodynamics of the model. We also analyse and discuss the existence of form-invariance dualities between solutions. A general way of defining the latter in an appropriate fashion for scalar fields is put forward.

Abelian symmetries in the two-Higgs-doublet model with fermions

We classify all possible implementations of an Abelian symmetry in the two-Higgs-doublet model with fermions. We identify those symmetries which are consistent with nonvanishing quark masses and a Cabibbo-Kobayashi-Maskawa quark-mixing matrix (CKM), which is not block-diagonal. Our analysis takes us from a plethora of possibilities down to 246 relevant cases, requiring only 34 distinct matrix forms. We show that applying Z(n) with n >= 4 to the scalar sector leads to a continuous U(1) symmetry in the whole Lagrangian. Finally, we address the possibilities of spontaneous CP violation and of natural suppression of the flavor-changing neutral currents. We explain why our work is relevant even for non-Abelian symmetries.

Leptonic mixing, family symmetries, and neutrino phenomenology

Tribimaximal leptonic mixing is a mass-independent mixing scheme consistent with the present solar and atmospheric neutrino data. By conveniently decomposing the effective neutrino mass matrix associated to it, we derive generic predictions in terms of the parameters governing the neutrino masses. We extend this phenomenological analysis to other mass-independent mixing schemes which are related to the tribimaximal form by a unitary transformation. We classify models that produce tribimaximal leptonic mixing through the group structure of their family symmetries in order to point out that there is often a direct connection between the group structure and the phenomenological analysis. The type of seesaw mechanism responsible for neutrino masses plays a role here, as it restricts the choices of family representations and affects the viability of leptogenesis. We also present a recipe to generalize a given tribimaximal model to an associated model with a different mass-independent mixing scheme, which preserves the connection between the group structure and phenomenology as in the original model. This procedure is explicitly illustrated by constructing toy models with the transpose tribimaximal, bimaximal, golden ratio, and hexagonal leptonic mixing patterns.

Anomaly-free U(1) gauge symmetries in neutrino seesaw flavor models

Agência Financiadora: Fundação para a Ciência e a Tecnologia (FCT) - PEst-OE/FIS/UI0777/2013; CERN/FP/123580/2011; PTDC/FIS-NUC/0548/2012

Sovereign debt and CRAs: Where does the power lie?

A Work Project, presented as part of the requirements for the Award of a Masters Degree in Management from the NOVA – School of Business and Economics

K-invariantes en el algebra universal de un grupo de Lie semisimple

Tarea (A):(...) Trataremos de extender a Sp(n,1) los resultados conseguidos sobre la imagen del homomorfismo de Lepowsky cuando G es SO(n,1) ó SU(n,1). (...) Tarea (B): (...) Para todo grupo de Lie de rango uno, con rango (G) = rango (K), los elementos del álgebra B son W-invariantes y que este resultado ya ha sido establecido para los grupos SO(2n,1) y SU(n,1); durante el período correspondiente a este subsidio esperamos extender este resultado a todo grupo de Lie de rango uno con rango (G) = rango (K). Tarea (C): Durante este período esperamos también avanzar en la determinación del dual unitario del grupo Spin (2n,C).

Representaciones de grupos y álgebras de Lie

Este proyecto cuenta con 7 subproyectos.Subproyecto: Restricciones de representaciones de cuadrado integrable.Se continuará trabajando en el problema de restringir representaciones de cuadrado integrable en un grupo de Lie a un subgrupo semisimple o la factor unipotente de un subgrupo parabólico. En particular, se continuará analizando el caso de restringir desde el grupo SO(2n,1) al subgrupo SO(2) x SO(2n-2,1) y al factor unipotente del parabólico minimal de un grupo de Lie clásico de rango uno. Subproyecto: Representación metapléctica y grupos de Heisenberg generalizados. Se estudia la restricción de la representación metapléctica a subgrupos del grupo metapléctico. Subproyecto: Álgebras de tipo H. Se estudiarán estructuras de biálgebra en las álgebras de tipo H, álgebras de Lie nilpotentes de dos etapas. Se continuará con el estudio de cuantizaciones de álgebras de tipo H. Se estudiarán propiedades geométricas de las funciones theta generalizadas que surgen de álgebras de tipo H. Subproyecto: Módulos de peso máximo. Se intenta dar una respuesta al problema de clasificación de módulos quasifinitos de peso máximo sobre ciertas álgebras de dimensión infinita. Subproyecto: Cuantización de las álgebras de tipo H. Se tratará de cuantizar las álgebras de tipo H, álgebras de Lie nilpotentes de dos etapas. Se trabajará con una definición más general de las álgebras de Heisenberg, tratando de encontrar teoremas tipo Stone-Von Neumann y generalizaciones de las funciones theta. Subproyecto: Continuación analítica de integrales de coeficientes matriciales. Se analiza la existencia de continuación holomorfa de la integral a lo largo de un grupo semisimple real de las potencias complejas de un coeficiente matricial de una representación irreducible admisible. Subproyecto: Cálculo explícito de soluciones fundamentales de operadores invariantes. Se analizan condiciones en el polinomio que define un operador diferencial k-invariante para que resulte hipoellítico. Se trata en particular el caso del grupo SO(n,1). Subproyecto 7: Generadores de Goldie. Se trata de encontrar algoritmos para el cálculo de generadores de Goldie.

Algebras de Lie de tipo H

Es mi intención centrar mis investigaciones en los próximos años en las álgebras de Lie tipo H. Es nuestro objetivo encontrar nuevas familias de álgebras regulares no de tipo H y verificar la existencia o no de irreducibles cumpliendo de estas propiedades. En particular es interesante plantear su cuantización, es decir encontrar estructuras de álgebras de Hopf que sean deformaciones del álgebra envolvente correspondiente al álgebra de Lie en estudio. En particular estudiaremos si existen cuantizaciones quasitriangulares lo que nos llevaría soluciones de la ecuación de Yang-Baxter cuántica. Hasta ahora hemos logrado la cuantización en ciertos casos particulares. Para comprender cómo deben ser hechas las cuantizaciones en forma más general es necesario realizar un estudio sistemático de las estructuras de la biálgebra de las álgebras de Lie de tipo H. En particular se tratarán de detectar estructuras de biálgebra quasitriangulares y por consiguientes soluciones de la ecuación de Yang-Baxter clásica. Es un resultado conocido que las funciones de theta se pueden expresar como coeficiente matricial de la representación de Stone-Von Neumann. De los teoremas de Stone-Von Neumann para álgebras de tipo H surgen entonces funciones que serían una generalización de las funciones theta; es nuestro objetivo encontrar propiedades de estas funciones que puedan ser de interés.

Estructura, representaciones y cohomología de álgebras de Lie nilpotents. Structure, represntations and cohomology of nilpotent Lie algebras.

El objetivo de este proyecto es obtener resultados de calidad en el área de las representaciones y cohomología de álgebras de Lie complejas nilpotentes de dimensión finita. Los objetivos específicos son (1) Demostrar que la familia de nilradicales parabólicos de las subálgebras de Lie semisimples satisfacen la conjetura del rango toral. (2) Calcular explícitamente la cohomología, aunque sea en grados bajos, de las álgebras de Lie 3-pasos nilpotentes libres y las álgebras $\mathfrak{gl}(2,A_{k})$ donde $A_{k}$ es el álgebra de quiver truncada en $k$ asociada a un quiver cíclico de $k$ flechas (y $k$ vértices). (3) Determinar explícitamente qué diagramas de Young aparecen en la cohomología, calculada por Kostant, de los nilradicales parabólicos de las subálgebras de Lie semisimples. (4) Mejorar las actuales cotas para las representaciones fieles de dimensión mínima de álgebras de Lie 3-pasos nilpotentes.

(When) would i lie to you? comment on? deception: the role of consequences?

This paper reconsiders the evidence on lying or deception presented in Gneezy (2005,American Economic Review). We argue that Gneezy?s data cannot reject the hipótesis that people are one of two kinds: either a person will never lie, or a person will lie whenever she prefers the outcome obtained by lying over the outcome obtained by telling the truth. This implies that so long as lying induces a preferred outcome over truth-telling, a person?s decisión of whether to lie may be completely insensitive to other changes in the induced outcomes, such as exactly how much she monetarily gains relative to how much she hurts an anonymous partner. We run new but similar experiments to those of Gneezy in order to test this hypothesis. We find that our data cannot reject this hypothesis either, but we also discover substantial differences in behavior between our sub jects and Gneezy?s sub jects.