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17 pages, 296 KiB

Open AccessArticle

Crossed Modules and Non-Abelian Extensions of Differential Leibniz Conformal Algebras

by Hui Wu, Shuangjian Guo and Xiaohui Zhang

Axioms 2024, 13(10), 685; https://doi.org/10.3390/axioms13100685 - 2 Oct 2024

Viewed by 387

In this paper, we introduce two-term differential

L e i b_{\infty}

-conformal algebras and give characterizations of some particular classes of such two-term differential

L e i b_{\infty}

-conformal algebras. Furthermore, we discuss the classification of the non-Abelian extensions in terms [...] Read more.

In this paper, we introduce two-term differential

L e i b_{\infty}

-conformal algebras and give characterizations of some particular classes of such two-term differential

L e i b_{\infty}

-conformal algebras. Furthermore, we discuss the classification of the non-Abelian extensions in terms of non-Abelian cohomology groups. Finally, we explore the inducibility of pairs of automorphisms and derive the analog Wells exact sequences under the circumstance of differential Leibniz conformal algebras. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

18 pages, 391 KiB

Open AccessArticle

Integral Quantization for the Discrete Cylinder

by Jean-Pierre Gazeau and Romain Murenzi

Quantum Rep. 2022, 4(4), 362-379; https://doi.org/10.3390/quantum4040026 - 21 Sep 2022

Cited by 5 | Viewed by 2269

Abstract

Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalized positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is their ability to circumvent problems due to the presence of singularities in the classical models. In this paper, we implement covariant integral quantizations for systems whose phase space is

Z \times S^{1}

, i.e., for systems moving on the circle. The symmetry group of this phase space is the discrete & compact version of the Weyl–Heisenberg group, namely the central extension of the abelian group

Z \times SO (2)

. In this regard, the phase space is viewed as the right coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on

L^{2} (S^{1})

, is square integrable on the phase space. We show how to derive corresponding covariant integral quantizations from (weight) functions on the phase space and resulting resolution of the identity. As particular cases of the latter we recover quantizations with de Bièvre-del Olmo–Gonzales and Kowalski–Rembielevski–Papaloucas coherent states on the circle. Another straightforward outcome of our approach is the Mukunda Wigner transform. We also look at the specific cases of coherent states built from shifted gaussians, Von Mises, Poisson, and Fejér kernels. Applications to stellar representations are in progress. Full article

25 pages, 383 KiB

Open AccessArticle

Riemann–Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: The Matrix Discrete Painlevé IV

by Amílcar Branquinho, Ana Foulquié Moreno, Assil Fradi and Manuel Mañas

Mathematics 2022, 10(8), 1205; https://doi.org/10.3390/math10081205 - 7 Apr 2022

Cited by 1 | Viewed by 1684

Abstract

In this paper, the Riemann–Hilbert problem, with a jump supported on an appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of the corresponding matrix biorthogonal polynomials associated with Laguerre type matrices of weights—which are constructed in terms of a given matrix Pearson equation. First and second order differential systems for the fundamental matrix, solution of the mentioned Riemann–Hilbert problem, are derived. An explicit and general example is presented to illustrate the theoretical results of the work. The non-Abelian extensions of a family of discrete Painlevé IV equations are discussed. Full article

(This article belongs to the Section Computational and Applied Mathematics)

20 pages, 344 KiB

Open AccessArticle

The Axial Anomaly in Lorentz Violating Theories: Towards the Electromagnetic Response of Weakly Tilted Weyl Semimetals

by Andrés Gómez and Luis Urrutia

Symmetry 2021, 13(7), 1181; https://doi.org/10.3390/sym13071181 - 30 Jun 2021

Cited by 4 | Viewed by 1759

Abstract

Using the path integral formulation in Euclidean space, we extended the calculation of the abelian chiral anomalies in the case of Lorentz violating theories by considering a new fermionic correction term provided by the standard model extension, which arises in the continuous Hamiltonian of a weakly tilted Weyl semimetal, and whose cones have opposite tilting. We found that this anomaly is insensitive to the tilting parameter, retaining its well-known covariant form. This independence on the Lorentz violating parameters is consistent with other findings reported in the literature. The initially imposed gauge invariant regularization was consistently recovered at the end of the calculation by the appearance of highly non-trivial combinations of the covariant derivatives, which ultimately managed to give only terms containing the electromagnetic tensor. We emphasize that the value of the anomaly with an arbitrary parameter is not automatically related to the effective action describing the electromagnetic response of such materials. Full article

(This article belongs to the Special Issue Space-Time Symmetries and Violations of Lorentz Invariance)

6 pages, 533 KiB

Open AccessEditorial

Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

by Frédéric Barbaresco and Jean-Pierre Gazeau

Entropy 2019, 21(3), 250; https://doi.org/10.3390/e21030250 - 6 Mar 2019

Cited by 1 | Viewed by 3764

Abstract

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

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10 pages, 306 KiB

Open AccessArticle

Modified Born–Infeld-Dilaton-Axion Coupling in Supersymmetry

by Yermek Aldabergenov and Sergei V. Ketov

Symmetry 2019, 11(1), 14; https://doi.org/10.3390/sym11010014 - 24 Dec 2018

Cited by 2 | Viewed by 2635

Abstract

We propose the supersymmetric extension of the modified Born–Infeld-axion-dilaton non-linear electrodynamics that has confined static abelian solutions used for describing the electromagnetic confinement in the presence of axion and dilaton fields, as well as charged matter. The supersymmetric extension also has the non-trivial [...] Read more.

(This article belongs to the Special Issue Supersymmetric Field Theory 2018)

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302 KiB

Open AccessArticle

Cohomology Characterizations of Diagonal Non-Abelian Extensions of Regular Hom-Lie Algebras

by Lina Song and Rong Tang

Symmetry 2017, 9(12), 297; https://doi.org/10.3390/sym9120297 - 5 Dec 2017

Viewed by 2763

Abstract

In this paper, first we show that under the assumption of the center of

h

being zero, diagonal non-abelian extensions of a regular Hom-Lie algebra

g

by a regular Hom-Lie algebra

h

are in one-to-one correspondence with Hom-Lie algebra morphisms from

g

to [...] Read more.

In this paper, first we show that under the assumption of the center of

h

being zero, diagonal non-abelian extensions of a regular Hom-Lie algebra

g

by a regular Hom-Lie algebra

h

are in one-to-one correspondence with Hom-Lie algebra morphisms from

g

Out (h)

. Then for a general Hom-Lie algebra morphism from

g

Out (h)

, we construct a cohomology class as the obstruction of existence of a non-abelian extension that induces the given Hom-Lie algebra morphism. Full article

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