Group theory and quasiprobability integrals of Wigner functions

The integral of the Wigner function of a quantum-mechanical system over a region or its boundary in the classical phase plane, is called a quasiprobability integral. Unlike a true probability integral, its value may lie outside the interval [0, 1]. It is characterized by a corresponding selfadjoint operator, to be called a region or contour operator as appropriate, which is determined by the characteristic function of that region or contour. The spectral problem is studied for commuting families of region and contour operators associated with concentric discs and circles of given radius a. Their respective eigenvalues are determined as functions of a, in terms of the Gauss-Laguerre polynomials. These polynomials provide a basis of vectors in a Hilbert space carrying the positive discrete series representation of the algebra su(1, 1) approximate to so(2, 1). The explicit relation between the spectra of operators associated with discs and circles with proportional radii, is given in terms of the discrete variable Meixner polynomials.

Pseudodifferential operators with C*-algebra-valued symbols: Abstract characterizations

Given a separable unital C*-algebra C with norm parallel to center dot parallel to, let E-n denote the Banach-space completion of the C-valued Schwartz space on R-n with norm parallel to f parallel to(2)=parallel to < f, f >parallel to(1/2), < f, g >=integral f(x)* g(x)dx. The assignment of the pseudodifferential operator A=a(x,D) with C-valued symbol a(x,xi) to each smooth function with bounded derivatives a is an element of B-C(R-2n) defines an injective mapping O, from B-C(R-2n) to the set H of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert module E-n. In this paper, we construct a left-inverse S for O and prove that S is injective if C is commutative. This generalizes Cordes' description of H in the scalar case. Combined with previous results of the second author, our main theorem implies that, given a skew-symmetric n x n matrix J and if C is commutative, then any A is an element of H which commutes with every pseudodifferential operator with symbol F(x+J xi), F is an element of B-C(R-n), is a pseudodifferential operator with symbol G(x - J xi), for some G is an element of B-C(R-n). That was conjectured by Rieffel.

Free field and parafermionic realizations of twisted su(3)(k)((2)) current algebra

Free field and twisted parafermionic representations of twisted su(3)(k)((2)) current algebra are obtained. The corresponding twisted Sugawara energy-momentum tensor is given in terms of three (beta, gamma) pairs and two scalar fields and also in terms of twisted parafermionic currents and one scalar field. Two screening currents of the first kind are presented in terms of the free fields.

The spectrum of an open vertex model based on the U(q)[SU(2)] algebra at roots of unity

We study the exact solution of an N-state vertex model based on the representation of the U(q)[SU(2)] algebra at roots of unity with diagonal open boundaries. We find that the respective reflection equation provides us one general class of diagonal K-matrices having one free-parameter. We determine the eigenvalues of the double-row transfer matrix and the respective Bethe ansatz equation within the algebraic Bethe ansatz framework. The structure of the Bethe ansatz equation combine a pseudomomenta function depending on a free-parameter with scattering phase-shifts that are fixed by the roots of unity and boundary variables. (C) 2010 Elsevier B.V. All rights reserved.

Polar orthogonal representations of real reductive algebraic groups

We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generalizes some results of the structural theory of real reductive Lie algebras. (c) 2008 Elsevier Inc. All rights reserved.

Partial projective representations and partial actions

The concept of a partial projective representation of a group is introduced and studied. The interaction with partial actions is explored. It is shown that the factor sets of partial projective representations over a field K are exactly the K-valued twistings of crossed products by partial actions. (C) 2009 Elsevier B.V. All rights reserved.

Free field realizations of the elliptic affine Lie algebra sl(2, R) circle plus (Omega(R)/dR)

In this paper we construct two free field realizations of the elliptic affine Lie algebra sl(2, R) circle plus Omega(R)/dR where R = C[t. t(-1), u vertical bar u(2) = t(3) - 2bt(2) + t]. The first realization provides an analogue of Wakimoto`s construction for Affine Kac-Moody algebras, but in the setting of the elliptic affine Lie algebra. The second realization gives new types of representations analogous to Imaginary Verma modules in the Affine setting. (c) 2009 Elsevier B.V. All rights reserved.

Hq(4) symmetry: the linear q-harmonic oscillator based on generalized irreps of the q-deformed Heisenberg algebra

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley

Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois' theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama's lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn's theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside's paqb Theorem; Injective modules.

Kleiner's theorem for unitary representations of posets

A subspace representation of a poset S = {s(1), ..., S-t} is given by a system (V; V-1, ..., V-t) consisting of a vector space V and its sub-spaces V-i such that V-i subset of V-j if s(i) (sic) S-j. For each real-valued vector chi = (chi(1), ..., chi(t)) with positive components, we define a unitary chi-representation of S as a system (U: U-1, ..., U-t) that consists of a unitary space U and its subspaces U-i such that U-i subset of U-j if S-i (sic) S-j and satisfies chi 1 P-1 + ... + chi P-t(t) = 1, in which P-i is the orthogonal projection onto U-i. We prove that S has a finite number of unitarily nonequivalent indecomposable chi-representations for each weight chi if and only if S has a finite number of nonequivalent indecomposable subspace representations; that is, if and only if S contains any of Kleiner's critical posets. (c) 2012 Elsevier Inc. All rights reserved.

Partial projective representations and partial actions II

This paper is a continuation of Dokuchaev and Novikov (2010) [8]. The interaction between partial projective representations and twisted partial actions of groups considered in Dokuchaev and Novikov (2010) [8] is treated now in a categorical language. In the case of a finite group G, a structural result on the domains of factor sets of partial projective representations of G is obtained in terms of elementary partial actions. For arbitrary G we study the component pM'(G) of totally-defined factor sets in the partial Schur multiplier pM(G) using the structure of Exel's semigroup. A complete characterization of the elements of pM'(G) is obtained for algebraically closed fields. (C) 2011 Elsevier B.V. All rights reserved.

Representations of semisimple Lie algebras

La tesi è dedicata allo studio delle rappresentazioni delle algebre di Lie semisemplici su un campo algebricamente chiuso di caratteristica zero. Mediante il teorema di Weyl sulla completa riducibilità, ogni rappresentazione di dimensione finita di una algebra di Lie semisemplice è scrivibile come somma diretta di sottorappresentazioni irriducibili. Questo permette di poter concentrare l'attenzione sullo studio delle rappresentazioni irriducibili. Inoltre, mediante il ricorso all'algebra inviluppante universale si ottiene che ogni rappresentazione irriducibile è una rappresentazione di peso più alto. Perciò è naturale chiedersi quando una rappresentazione di peso più alto sia di dimensione finita ottenendo che condizione necessaria e sufficiente perché una rappresentazione di peso più alto sia di dimensione finita è che il peso più alto sia dominante. Immediata è quindi l'applicazione della teoria delle rappresentazioni delle algebre di Lie semisemplici nello studio delle superalgebre di Lie, in quanto costituite da un'algebra di Lie e da una sua rappresentazione, dove viene utilizzata la tecnica della Z-graduazione che viene utilizzata per la prima volta da Victor Kac nello studio delle algebre di Lie di dimensione infinita nell'articolo ''Simple irreducible graded Lie algebras of finite growth'' del 1968.

Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.

Explicit description of generic representations for quivers of type An or Dn

We describe explicitly a generic representation for Dynkin quivers of type An or Dn for any dimension vector.