Killing Form Research Articles

Gender-Based Violence in Some Pentecostal Churches—A South African Study

One of the most devastating practices to be normalised in modern times is gender-based violence. Women and girls in South Africa can no longer enjoy the freedom that was promised by the country’s Freedom Charter during the dawn of democracy. Rape, killing and other forms of female persecution have become common practices, and the most worrying part of this is that the perpetrators of this violence seem to enjoy more freedom than their victims. Unfortunately, this kind of violence is no longer a secular issue but something that is now encountered in religious circles, places in which most people would hope to be sheltered and protected. The article investigates these kinds of violence within the parameters of religious institutions, specifically Pentecostal churches, and also makes some suggestions as to what kinds of approaches we need as a country to eliminate this pandemic. This article uses a desktop/secondary approach to gather data and to reach the conclusions made as a bases for the argument.

Outer billiards in the spaces of oriented geodesics of the three‐dimensional space forms

AbstractLet be the three‐dimensional space form of constant curvature , that is, Euclidean space , the sphere , or hyperbolic space . Let be a smooth, closed, strictly convex surface in . We define an outer billiard map on the four‐dimensional space of oriented complete geodesics of , for which the billiard table is the subset of consisting of all oriented geodesics not intersecting . We show that is a diffeomorphism when is quadratically convex. For , has a Kähler structure associated with the Killing form of . We prove that is a symplectomorphism with respect to its fundamental form and that can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in defined in terms of the standard symplectic structure. We show that does not preserve the fundamental symplectic form on associated with the cross product on , for . We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.

The minimal number of homogeneous geodesics depending on the signature of the Killing form

The minimal number of homogeneous geodesics depending on the signature of the Killing form

Mostow's decomposition theorem for L⁎-groups and applications to affine coadjoint orbits and stable manifolds

Mostow's decomposition theorem is a refinement of the polar decomposition. It states the following. Let G be a compact connected semi-simple Lie group with Lie algebra g. Given a subspace h of g such that [X,[X,Y]]∈h for all X, Y in h, the complexified group GC is homeomorphic to the product G⋅exp⁡im⋅exp⁡ih, where m is the orthogonal of h in g with respect to the Killing form. This Theorem is related to geometric properties of the non-positively curved space of positive-definite symmetric matrices and to a characterization of its geodesic subspaces. The original proof of this Theorem given by Mostow in [16] uses the compactness of G. We give a proof of this theorem using the completeness of the Lie algebra g instead, which can therefore be applied to an L⁎-group of arbitrary dimension. A different proof has been obtained independently by G. Larotonda in [12,13]. Important applications of this theorem to the geometry of (affine) coadjoint orbits and stable manifolds are given in the present paper.

Drinfeld realization of the centrally extended psl(2|2) Yangian algebra with the manifest coproducts

The Lie superalgebra psl(2|2) is recognized as a quite special algebra. In mathematics, it has the vanishing Killing form and allows for the three-dimensional central extension. In physics, it shows up as the symmetry of the one-dimensional Hubbard model and the asymptotic S-matrix of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. In this paper, we present the Drinfeld realization of the Yangian algebra associated with the centrally extended Lie superalgebra psl(2|2). Furthermore, we show that it possesses Hopf algebra structures, particularly the coproducts. The idea to prove the existence of manifest coproducts is as follows. First, we shall introduce them to Levendorskii’s realization, a system of a finite truncation of Drinfeld generators. Second, we show that Levendorskii’s realization is isomorphic to the Drinfeld realization by induction.

Universal 1-loop divergences for integrable sigma models

We present a simple, new method for the 1-loop renormalization of integrable σ-models. By treating equations of motion and Bianchi identities on an equal footing, we derive ‘universal’ formulae for the 1-loop on-shell divergences, generalizing case-by-case computations in the literature. Given a choice of poles for the classical Lax connection, the divergences take a theory-independent form in terms of the Lax currents (the residues of the poles), assuming a ‘completeness’ condition on the zero-curvature equations. We compute these divergences for a large class of theories with simple poles in the Lax connection. We also show that ℤT coset models of ‘pure-spinor’ type and their recently constructed η- and λ-deformations are 1-loop renormalizable, and 1-loop scale-invariant when the Killing form vanishes.

A class of extended Lie superalgebras and their applications

We construct a class of extended Lie superalgebras, including a generalized Lie superalgebra B(0,1), a generalized Lie superalgebra spl(2,1), a generalized Lie superalgebra spo(2,2), and a generalized Lie superalgebra osp(2,2). Considering the applications of these Lie superalgebras, we obtain three different nonisospectral super AKNS hierarchies. It follows that many classical and new equations are deduced by discussing some reductions of these hierarchies, such as the nonlinear Schrödinger (NLS) equation, the Burgers equation, the heat equation, the Fokker–Planck equation, the KdV equation, and so on. Based on the Killing form on these extended Lie superalgebras, we obtain the super Hamiltonian structures of these three nonisospectral super AKNS hierarchies.

Einstein warped product spaces on Lie groups

We consider a compact Lie group with bi-invariant metric, coming from the Killing form. In this paper, we study Einstein warped product space, $M = M_1 \times_{f_1} M_2$ for the cases, $(i)$ $M_1$ is a Lie group $(ii)$ $M_2$ is a Lie group and $(iii)$ both $M_1$ and $M_2$ are Lie groups. Moreover, we obtain the conditions for an Einstein warped product of Lie groups to become a simple product manifold. Then, we characterize the warping function for generalized Robertson-Walker spacetime, $(M = I \times_{f_1} G_2, - dt^2 + f_1^2 g_2)$ whose fiber $G_2$, being semi-simple compact Lie group of $\dim G_2>2$, having bi-invariant metric, coming from the Killing form.

Positive Hermitian curvature flow on special linear groups and perfect solitons

We study invariant solutions to the positive Hermitian curvature flow, introduced by Ustinovskiy, on complex Lie groups. We show in particular that the canonical scale-static metrics on the special linear groups, arising from the Killing form, are dynamically unstable. This disproves a conjecture of Ustinovskiy. We also construct certain perfect Lie groups that admit at least two distinct invariant solitons for the flow, only one of which is algebraic. This is the second known example of a geometric flow with non-algebraic, homogeneous solitons. The first being the G2-Laplacian flow.

Conformal Killing forms in Kähler geometry

For Kähler manifolds, we explicitly determine the solution to the conformal Killing form equation in middle degree. In particular, we complete the classification of conformal Killing forms on compact Kähler manifolds. We give the first examples of conformal Killing forms on Kähler manifolds not coming from Hamiltonian 2-forms. These are supported by Calabi-type manifolds over a Kähler Einstein base. In this setup, we also give structure results and examples for the closely related class of Hermitian Killing forms.

Global Stability of the Pluriclosed Flow on Compact Simply Connected Simple Lie Groups of Rank Two

AbstractWe compute the (1,1)-Aeppli cohomology of compact simply connected simple Lie groups of rank two. In particular, we verify that they are of dimension one and generated by the classes of the Bismut flat metrics coming from the Killing forms. This yields a result on the stability of the pluriclosed flow on these manifolds. Moreover, we show that for compact simply connected simple Lie groups of rank two the Dolbeaut cohomology, as well as the Bott–Chern and the Aeppli cohomologies, arises from just the left-invariant forms and we computed the whole Bott–Chern diamonds of SU(3) and Spin(5) when they are equipped with a left-invariant isotropic complex structure.

Economic Implications of Trade in Wildlife: From Legality to Illegality

Illegal trade in wildlife has caused the decline of many species in Nigeria, Africa and the world. Illegal killing, smuggling and other forms of illicit trade in wildlife do not only hurt the economy and the ecosystem, but they also fuel organised crime as well as feed corruption and insecurity, undermining the sustainable development of the country. Wildlife in Nigeria is under pressure as most endangered species, are traded both domestically and internationally. Nigeria is not only a source of wildlife products but has also emerged as a major transit country for wildlife trafficking according to the current World wildlife Crime Report. Using a literature-based review, this paper critically examines the legality and illegality of wildlife trade in Nigeria in line with the provisions of extant laws and how this scenario has grossly hampered global efforts toward combating wildlife trafficking. It proffers amendment of the text of extant laws to reflect the intents and global aspirations to curb this scourge and preserve biodiversity for posterity. It further advocates global partnerships efforts to strengthen enforcement capacity to investigate and prosecute offenders.

Invariants from the Sweedler power maps on integrals

For a finite-dimensional Hopf algebra A with a nonzero left integral Λ, we investigate a relationship between Pn(Λ) and PnJ(Λ), where Pn and PnJ are respectively the n-th Sweedler power maps of A and the twisted Hopf algebra AJ. We use this relation to give several invariants of the representation category Rep(A) considered as a tensor category. As applications, we distinguish the representation categories of 12-dimensional pointed nonsemisimple Hopf algebras. Also, these invariants are sufficient to distinguish the representation categories Rep(K8), Rep(kQ8) and Rep(kD4), although they have been completely distinguished by their Frobenius-Schur indicators. We further reveal a relationship between the right integrals λ in A⁎ and λJ in (AJ)⁎. This can be used to give a uniform proof of the remarkable result which says that the n-th indicator νn(A) is a gauge invariant of A for any n∈Z. We also use the expression for λJ to give an alternative proof of the known result that the Killing form of the Hopf algebra A is invariant under twisting. As a result, the dimension of the Killing radical of A is a gauge invariant of A.

Dynamics of some new solutions to the coupled DSW equations traveling horizontally on the seabed

The system of (1+1)-coupled Drinfel’d–Sokolov–Wilson equations describes the surface gravity waves travelling horizontally on the seabed. The objective of the present research is to construct a new variety of analytical solutions for the system. The invariants are derived with the aid of Killing form by using the optimal algebra classification via Lie symmetry approach. The invariant solutions involve time, space variables, and arbitrary constants. Imposing adequate constraints on arbitrary constants, solutions are represented graphically to make them more applicable in designing sea models. The behavior of solutions shows asymptotic, bell-shaped, bright and dark soliton, bright soliton, parabolic, bright and kink, kink, and periodic nature. The constructed results are novel as the reported results [26,28,29,30,33,38,42,49] can be deduced from the results derived in this study. The remaining solutions derived in this study, are absolutely different from the earlier findings. In this study, the physical character of analytical solutions of the system could aid coastal engineers in creating models of beaches and ports.

Involutions of sl(2,k) and non-split, three-dimensional simple Lie algebras

We give a process to construct non-split, three-dimensional simple Lie algebras from involutions of [Formula: see text], where [Formula: see text] is a field of characteristic not two. Up to equivalence, non-split three-dimensional simple Lie algebras obtained in this way are parametrized by a subgroup of the Brauer group of [Formula: see text] and are characterized by the fact that their Killing form represents [Formula: see text]. Over local and global fields we re-express this condition in terms of Hilbert and Legendre Symbols and give examples of three-dimensional simple Lie algebras which can and cannot be obtained by this construction over the field of rationals.

On the Faulkner construction for generalized Jordan superpairs

In this paper, the well-known Faulkner construction is revisited and adapted to include the super case, which gives a bijective correspondence between generalized Jordan (super)pairs and faithful Lie (super)algebra (super)modules, under certain constraints (bilinear forms with properties analogous to the ones of a Killing form are required, and only finite-dimensional objects are considered). We always assume that the base field has characteristic different from 2.It is also proven that associated objects in this Faulkner correspondence have isomorphic automorphism group schemes. Finally, this correspondence will be used to transfer the construction of the tensor product to the class of generalized Jordan (super)pairs with “good” bilinear forms.

Dynamical behavior of similarity solutions of CKOEs with conservation law

The propagation of a current-fed string in 3-dimensional space interacting with an external magnetic field is described by the system of coupled Konno–Oono equations. Present research deals with the formation of one, two, and three dimensional optimal sub-algebras of the system. Then invariants are decided by using the Killing form. Similarity reductions and their solutions are determined for each sub-algebra. For strong integrability of the system, conserved vectors are also attained by using Noether’s theorem. The novelty of analytical solutions can be demonstrated by the fact that in earlier studies, all researchers solved coupled Konno–Oono equations involving only two components of the water wave, but in this study, the authors solved its more general form. Some comments are also made to demonstrate that the results in this article are preferable to those in previous studies. To make the solutions physically meaningful, they are reinforced with numerical simulation. Periodic, single soliton, doubly bright, bright and dark multisolitons, asymptotic, travelling waves with dark soliton, progressive, highly progressive, and stationary solutions can be seen in the profiles. Such solutions can be beneficial in optical nonlinearity, electro-magnetics, and quantum fields.

Optimal Subalgebra of GKP by Using Killing Form, Conservation Law and Some More Solutions

Optimal Subalgebra of GKP by Using Killing Form, Conservation Law and Some More Solutions

Generalized self-duality for the Yang-Mills-Higgs system

Self-duality is a very important concept in the study and applications of topological solitons in many areas of Physics. The rich mathematical structures underlying it lead, in many cases, to the development of exact and non-perturbative methods. We present a generalization of the Yang-Mills-Higgs system by the introduction of scalar fields assembled in a symmetric and invertible matrix h of the same dimension as the gauge group. The coupling of such new fields to the gauge and Higgs fields is made by replacing the Killing form, in the contraction of the group indices, by the matrix h in the kinetic term for the gauge fields, and by its inverse in the Higgs field kinetic term. The theory is conformally invariant in the three dimensional space R^3. An important aspect of the model is that for practically all configurations of the gauge and Higgs fields the new scalar fields adjust themselves to solve the modified self-duality equations. We construct solutions using a spherically symmetric ans\"atz and show that the 't Hooft-Polyakov monopole becomes a self-dual solution of such modified Yang-Mills-Higgs system. We use an ans\"atz based on the conformal symmetry to construct vacuum solutions presenting non-trivial toroidal magnetic fields.

Conformal Killing forms on 2-step nilpotent Riemannian Lie groups

Abstract We study left-invariant conformal Killing 2- or 3-forms on simply connected 2-step nilpotent Riemannian Lie groups. We show that if the center of the group is of dimension greater than or equal to 4, then every such form is automatically coclosed (i.e. it is a Killing form). In addition, we prove that the only Riemannian 2-step nilpotent Lie groups with center of dimension at most 3 and admitting left-invariant non-coclosed conformal Killing 2- and 3-forms are the following: The Heisenberg Lie groups and their trivial 1-dimensional extensions, endowed with any left-invariant metric, and the simply connected Lie group corresponding to the free 2-step nilpotent Lie algebra on 3 generators, with a particular 1-parameter family of metrics. The explicit description of the space of conformal Killing 2- and 3-forms is provided in each case.