108,210 research outputs found
Álgebras de Hopf trançadas
No full textDissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Programa de Pós-Graduação em Matemática Pura e Aplicada, Florianópolis, 2013Álgebras de Nichols são ferramentas importantes para a classificação de álgebras de Hopf pontuadas (veja [3]). Uma álgebra de Nichols é, em suma, uma álgebra de Hopf trançada e graduada. Ao considerarmos a categoria dos módulos de Yetter-Drinfeld sobre uma álgebra de Hopf com antípoda bijetora, cria-se o ambiente para definir álgebras de Hopf trançadas nessa categoria (o que pode ser feito em uma categoria trançada qualquer). Esse trabalho desenvolve esse problema, isto é, dada uma álgebra de Hopf H com antípoda bijetora sobre um corpo k, nossos principais objetivos são estudar álgebras de Hopf trançadas na categoria dos módulos de Yetter-Drinfeld sobre H e mostrar a existência e a unicidade da álgebra de Nichols de um módulo de Yetter-Drinfeld sobre H.Abstract : Nichols algebras play an important role to classify pointed Hopf algebras. If we consider the category of modules of Yetter-Drinfeld over a Hopf algebra H with bijective antipode, we get a braided category and so it is possible to define a braided Hopf algebra there. In this work, we consider this kind of problem, i. e., given a Hopf algebra H with bijective antipode (over a field k), we consider the category of modules of Yetter-Drinfeld over it. We study braided Hopf algebras in this category and also we prove the existence and uniqueness of the Nichols algebra of a Yetter-Drinfeld module
Extensões de álgebras obtidas a partir de álgebras de Hopf
No full textDissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Programa de Pós-graduação em Matemática e Computação Científica, Florianópolis, 2011Neste trabalho fazemos uma descrição completa do grupo quântico A(SL_q(2)), em que q é a raiz cúbica da unidade, como uma extensão de Hopf-Galois fielmente plana de A(SL(2,C)) a partir da sequência exata de álgebras de Hopf A(SL(2,C)) A(SL_q(2)) A(F) determinada pelo morfismo de Frobenius Fr. Além disso, estendemos o resultado para o subgrupo quântico de Borel, obtendo a estrutura de produto cruzado. No mais, é feito um estudo dos resultados da teoria de álgebras de Hopf e da teoria de extensões de álgebras obtidas a partir de álgebras de Hopf. Ainda, mostramos que toda biálgebra que admite uma extensão de Hopf-Galois fielmente plana é uma álgebra de Hopf
Fermionic quantization of Hopf solitons
No full textIn this paper we show how to quantize Hopf solitons using the Finkelstein-Rubinstein approach. Hopf solitons can be quantized as fermions if their Hopf charge is odd. Symmetries of classical minimal energy configurations induce loops in configuration space which give rise to constraints on the wave function. These constraints depend on whether the given loop is contractible. Our method is to exploit the relationship between the configuration spaces of the Faddeev-Hopf and Skyrme models provided by the Hopf fibration. We then use recent results in the Skyrme model to determine whether loops are contractible. We discuss possible quantum ground states up to Hopf charge Q=7
On the Cohomology of Modules over Hopf Algebras
No full textAbstractLet R be a commutative ring. Define an FH-algebra H to be a Hopf algebra and a Frobenius algebra over R with a Frobenius homomorphism ψ such that ∑(h) h(1) ψ(h(2)) = ψ(h) · 1 for all h ϵ H. This is essentially the same as to consider finitely generated projective Hopf algebras with antipode. For modules over FH-algebras we develop a cohomology theory which is a generalization of the cohomology of finite groups. It generalizes also the cohomology of finite-dimensional restricted Lie algebras. In particular the following results are shown. The complete homology can be described in terms of the complete cohomology. There is a cup-product for the complete cohomology and some of the theorems for periodic cohomology of finite groups can be generalized. We also prove a duality theorem which expresses the cohomology of the “dual” of an H-module as the “dual” of the cohomology of the module. The last section provides techniques to describe under certain conditions the cohomology of H by the cohomology of sub- and quotient-algebras of H. In particular we have a generalization of the Hochschild-Serre spectral sequence for the cohomology of groups
When Hopf Algebras are Frobenius Algebras
No full textAbstractR. Larson and M. Sweedler recently proved that for free finitely generated Hopf algebras H over a principal ideal domain R the following are equivalent: (a) H has an antipode and (b) H has a nonsingular left integral. In this paper I give a generalization of this result which needs only a minor restriction, which, for example, always holds if pic(R) = 0 for the base ring R. A finitely generated projective Hopf algebra H over R has an antipode if and only if H is a Frobenius algebra with a Frobenius homomorphism ψ such that Σ h(1) ψ(h(2)) = ψ(h) · 1 for all h ϵ H. We also show that the antipode is bijective and that the ideal of left integrals is a free rank 1, R-direct summand of H
Álgebras de Hopf associadas a grafos tipo árvore
No full textDissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas, Programa de Pós-Graduação em Matemática e Computação Científica, Florianópolis, 2015.O presente trabalho tem como objetivo explorar algumas álgebras de Hopf construídas a partir de grafos do tipo árvore (com raiz). Estudam-se as álgebras de Connes-Kreimer e a de Grossman-Larson, e busca-se uma relação entre essas duas álgebras de Hopf. A relação encontrada é a dualidade separante. Também explora-se uma versão para árvores ordenadas das álgebras de Connes-Kreimer e de Grossman-Larson. Prova-se a dualidade dessas duas álgebras seguindo a ideia do caso anterior, mas não se consegue obter que a dualidade é separante. Um contraexemplo para isso é mostrado. Os capítulos iniciais apresentam a teoria básica de álgebras de Hopf e de Lie necessária para a leitura deste trabalho. Alguns resultados sobre biálgebras conexas com filtração e com graduação são vistos no capítulo 4, incluindo a demonstração do teorema de Milnor-Moore. O capítulo 5 apresenta as árvores (não-ordenadas e ordenadas) e as álgebras de Connes-Kreimer e de Grossman-Larson obtidas a partir das mesmas. Termina-se apresentando o teorema de Panaite e a dualidade entre essas duas álgebras de Hopf.Abstract : The present work explores some Hopf algebras built over rooted trees. The Hopf algebras of Connes-Kreimer and Grossman-Larson are studied, and a relationship between these algebras is investigated. The relationship between these algebras turns out to be a separating duality. A version of the Connes-Kreimer and Grossman-Larson algebras using ordered rooted trees is also investigated. A duality between these algebras is obtained, in the same way as the non-ordered case. However, it is not a separating duality, and a counterexample is shown.The first three chapters present the basic theory of Hopf algebras and Lie algebras required for the remainder of the work. Some results concerning gradded and filtered connected bialgebras are shown in chapter 4, including the proof of the Milnor-Moore theorem. The chapter 5 presents (non-ordered and ordered) rooted trees and the Connes-Kreimer and Grossman-Larson algebras built over them. A duality between these two algebras is, then, shown, by means of Panaite's theorem
Fredholm factorization of Wiener-Hopf scalar and matrix kernels
No full textA general theory to factorize the Wiener-Hopf (W-H) kernel using Fredholm Integral Equations (FIE) of the second kind is presented. This technique, hereafter called Fredholm factorization, factorizes the W-H kernel using simple numerical quadrature. W-H kernels can be either of scalar form or of matrix form with arbitrary dimensions. The kernel spectrum can be continuous (with branch points), discrete (with poles), or mixed (with branch points and poles). In order to validate the proposed method, rational matrix kernels in particular are studied since they admit exact closed form factorization. In the appendix a new analytical method to factorize rational matrix kernels is also described. The Fredholm factorization is discussed in detail, supplying several numerical tests. Physical aspects are also illustrated in the framework of scattering problems: in particular, diffraction problems. Mathematical proofs are reported in the pape
Hopf spaces /
No full textIncludes bibliographical references (pages 211-218) and index.Print version record.Front Cover; Hopf Spaces; Copyright Page; Table of Contents; Introduction; Chapter 0. Notations, conventions and preliminary observations; 0.1 Spaces and maps; 0.2 Homotopies; 0.3 Categories and adjoint maps; 0.4 Pullbacks, pushouts and Eckmann-Hilton duality; 0.5 O-spectra, ring spectra, generalized cohomology; Chapter I. The category of H-spaces; Introduction; 1.1 Basic properties of H-spaces; 1.2 Some special classes of H-spaces; 1.3 The structure of [ , H-space]; 1.4 H-deviation and H-homotopy equivalence; 1.5 Change of H-structures and H-maps; Chapter II. Homotopy properties of H-spacesIntroduction2.1 H-spaces and fibrations; 2.2 H-liftings; 2.3 Postnikov systems; 2.4 Actions, H-actions and principal fibrations; 2.5 HA and HC obstructions; 2.6 Homotopy solvability and homotopy nilpotency; Chapter III. The cohomology of H-spaces; Introduction; 3.1 The Hopf algebra H*(X,Zp ); 3.2 Some relations between the algebra H*(X,Zp) and the coalgebra H*( OX, Zp ); 3.3 Browder's Bockstein spectral sequence; 3.4 High order operations; Chapter IV. Mod p theory of H-spaces; Introduction; 4.1 p-equivalence and p-universal spaces; 4.2 mod p-homotopy; 4.3 Decomposition of o-equivalences4.4 A study of Ho spaces4.5 Mod P1 H-spaces; 4.6 The genus of an H-space; 4.7 Mixing homotopy types; 4.8 The non classical H-spaces and other applications; Chapter V. Non stable BP resolutions; Introduction; 5.1 Killing homology p-torsion; 5.2 Wilson's B(n,p)'s; 5.3 The groups [ , B(n,p)]; 5.4 H-maps into B(n,p); 5.5 Examples: Some properties of BU; 5.6 Non stable BP Adams resolutions; 5.7 Some simple applications; Bibliography; List of symbols; Index of terminologyElsevie
Subcritical Hopf bifurcations in a car-following model with reaction-time delay
No full textA nonlinear car-following model of highway traffic is considered, which includes the reaction-time delay of drivers. Linear stability analysis shows that the uniform flow equilibrium of the system loses its stability via Hopf bifurcations and thus oscillations can appear. The stability and amplitudes of the oscillations are determined with the help of normal-form calculations of the Hopf bifurcation that also handles the essential translational symmetry of the system. We show that the subcritical case of the Hopf bifurcation occurs robustly, which indicates the possibility of bistability. We also show how these oscillations lead to spatial wave formation as can be observed in real-world traffic flow
The Wiener-Hopf solution of the isotropic penetrable wedge problem: diffraction and total field
No full textThe diffraction of an incident plane wave by an isotropic penetrable wedge is studied using generalized Wiener-Hopf equations, and the solution is obtained using analytical and numerical-analytical approaches that reduce the Wiener-Hopf factorization to Fredholm integral equations of second kind. Mathematical aspects are described in a unified and consistent theory for angular region problems. The formulation is presented in the general case of skew incidence and several numerical tests at normal incidence are reported to validate the new technique. The solutions consider engineering applications in terms of GTD/UTD diffraction coefficients and total field
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