Double Extension Research Articles

On a remarkable class of flat pseudo-Riemannian Lie groups

A flat pseudo-Riemannian Lie group (G,μ) is a Lie group endowed with a flat left-invariant pseudo-Riemannian metric μ. Milnor showed that if (G,μ) is a flat Riemannian Lie group then ∇[g,g]=0, where ∇ is the Levi-Civita connection and [g,g] is the derived ideal of its Lie algebra g. In this paper, we study the class C of flat pseudo-Riemannian Lie groups which satisfy the property ∇[g,g]=0. We start by showing that the class C is rich of examples; for instance, C contains all flat Lorentzian nilpotent lie groups and all flat pseudo-Riemannian Lie groups such that the restriction of the metric to [g,g] is positive or negative definite. We characterize (G,μ)∈C such that [g,g] is nondegenerate for any signature. In particular, we show that the metric in this case is complete and the Lie algebra is unimodular. We prove that if (G,μ)∈C is a flat Lorentzian Lie group with degenerate derived ideal, then its Lie algebra is obtained by the double extension process, and the metric in this case may be incomplete. As examples, we classify such lie algebras up to dimension 4. We show that if (G,μ)∈C is nilpotent then its Lie algebra is obtained by a sequence of double extension of elements of C starting from an Euclidean abelian Lie algebra. If G is 2-step nilpotent, then we prove that (G,μ)∈C if and only if [g,g] is totally isotropic, and in particular dim⁡[g,g]≤ν where ν is the index of the metric. It is known that the Heisenberg Lie group H2k+1 admits a flat left-invariant pseudo-Riemannian metric if and only if k=1. We show that, if n,k,ν∈N⁎ such that 2ν≤n+2k+1 then, Rn×H2k+1 admits a flat left-invariant pseudo-Riemannian metric μ of index ν if and only if k≤ν+12. In particular we prove that (Rn×H2k+1,μ) must be in the class C.

The Structure of Geodesic Orbit Lorentz Nilmanifolds

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases. Those classes include weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz manifolds. Here, we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifolds. Those are the geodesic orbit Lorentz manifolds $$M = G/H$$ such that a nilpotent analytic subgroup of G is transitive on M. Suppose that there is a reductive decomposition $${\mathfrak {g}}= {\mathfrak {h}}\oplus {\mathfrak {n}}$$ (vector space direct sum) with $${\mathfrak {n}}$$ nilpotent. When the metric is nondegenerate on $$[{\mathfrak {n}},{\mathfrak {n}}]$$ , we show that $${\mathfrak {n}}$$ is abelian or 2-step nilpotent (this is the same result as for geodesic orbit Riemannian nilmanifolds), and when the metric is degenerate on $$[{\mathfrak {n}},{\mathfrak {n}}]$$ , we show that $${\mathfrak {n}}$$ is a Lorentz double extension corresponding to a geodesic orbit Riemannian nilmanifold. In the latter case, we construct examples to show that the number of nilpotency steps is unbounded.

Lie algebraic Carroll/Galilei duality

We characterize Lie groups with bi-invariant bargmannian, galilean, or carrollian structures. Localizing at the identity, we show that Lie algebras with ad-invariant bargmannian, carrollian, or galilean structures are actually determined by the same data: a metric Lie algebra with a skew-symmetric derivation. This is the same data defining a one-dimensional double extension of the metric Lie algebra and, indeed, bargmannian Lie algebras coincide with such double extensions, containing carrollian Lie algebras as an ideal and projecting to galilean Lie algebras. This sets up a canonical correspondence between carrollian and galilean Lie algebras mediated by bargmannian Lie algebras. This reformulation allows us to use the structure theory of metric Lie algebras to give a list of bargmannian, carrollian, and galilean Lie algebras in the positive-semidefinite case. We also characterize Lie groups admitting a bi-invariant (ambient) leibnizian structure. Leibnizian Lie algebras extend the class of bargmannian Lie algebras and also set up a non-canonical correspondence between carrollian and galilean Lie algebras.

Extensions, crossed modules and pseudo quadratic Lie type superalgebras

Extensions and crossed modules of Lie type superalgebras are introduced and studied. We construct homology and cohomology theories of Lie-type superalgebras. The notion of left super-invariance for a bilinear form is defined and we consider Lie type superalgebras endowed with nondegenerate, supersymmetric and left super-invariant bilinear form. Such Lie type superalgebras are called pseudo quadratic Lie type superalgebras. We show that any pseudo quadratic Lie type superalgebra induces a Jacobi-Jordan superalgebra. By using the method of double extension, we study pseudo quadratic Lie type superalgebras and theirs associated Jacobi-Jordan superalgebras.

Indecomposable doubling for representations of the type I Lie superalgebras sl(m/n) and osp(2/2n)

We establish that for the type I Lie superalgebras and , each Kac module admits a 1 parameter family of indecomposable double extensions. The result follows from the explicit evaluation of the H 1 Lie superalgebra cohomology valued in the tensor product of the module and its dual.

Double and Lagrangian extensions for quasi-Frobenius Lie superalgebras

A Lie superalgebra is called quasi-Frobenius if it admits a closed anti-symmetric non-degenerate bilinear form. We study the notion of double extensions of quasi-Frobenius Lie superalgebra when the form is either orthosymplectic or periplectic. We show that every quasi-Frobenius Lie superalgebra that satisfies certain conditions can be obtained as a double extension of a smaller quasi-Frobenius Lie superalgebra. We classify all 4-dimensional quasi-Frobenius Lie superalgebras, and show that such Lie superalgebras must be solvable. We study the notion of [Formula: see text]-extensions (or Lagrangian extensions) of Lie superalgebras, and show that they are classified by a certain cohomology space we introduce. Several examples are provided to illustrate our construction.

Equivalent constructions of nilpotent quadratic Lie algebras

The double extension and the T⁎-extension are classical methods for constructing finite dimensional quadratic Lie algebras. The first one gives an inductive classification in characteristic zero, while the latest produces quadratic non-associative algebras (not only Lie) out of arbitrary ones in characteristic different from 2. The classification of quadratic nilpotent Lie algebras can also be reduced to the study of free nilpotent Lie algebras and their invariant forms. In this work we will establish an equivalent characterization among these three construction methods. This equivalence reduces the classification of quadratic 2-step nilpotent to that of trivectors in a natural way. In addition, theoretical results will provide simple rules for switching among them.

Manin triples and non-degenerate anti-symmetric bilinear forms on Lie superalgebras in characteristic 2

In this paper, we introduce and develop the notion of a Manin triple for a Lie superalgebra g defined over a field of characteristic p=2. We find cohomological necessary conditions for the pair (g,g⁎) to form a Manin triple. We introduce the concept of Lie bi-superalgebras for p=2 and establish a link between Manin triples and Lie bi-superalgebras. In particular, we study Manin triples defined by a classical r-matrix with an extra condition (called an admissible classical r-matrix). A particular case is examined where g has an even invariant non-degenerate bilinear form. In this case, admissible r-matrices can be obtained inductively through the process of double extensions. In addition, we introduce the notion of double extensions of Manin triples, and show how to get a new Manin triple from an existing one.

Extended Double Bond Conjugation in the Chalcone Framework Favours MAO-B Inhibition: A Structural Perspective on Molecular Dynamics.

The monotropic membrane protein monoamine oxidase B (MAO-B) has been shown to be a crucial drug target for the treatment of neurodegenerative diseases. The design of recent inhibitor therapeutic agents of MAO-B involves conjugation and modification of a chalcone scaffold comprising two aryl or heteroaryl rings connected via a short spacer unit with rotatable bonds. Supported by experimental data, these modifications often result in high potent inhibitor compounds. In this study, we employ molecular dynamics simulations to unravel the impact of extended double bond conjugation in two novel compounds, F1 and MO10, toward the inhibition of the MAO-B protein. It was revealed that extended double bond conjugation induced a unidirectional orientation and motion of F1 and MO10, suggesting a stable binding pocket anchorage favouring high-affinity pocket interactions. Conformational analyses also revealed that the incorporated double bond extension impeded the motion of individual binding pocket residues, which subsequently disrupted the functionality of MAO-B. Real-time structural dynamics also revealed that the extended double bond conjugation mediated peculiar interactions with MAO-B binding pocket residues characterized by π-alkyl, π-π stacking, and π-sulphur interactions which buried both compounds into the hydrophobic core of MAO-B and ultimately induced higher binding affinities of both F1 and MO10. These insights present useful structural perspectives of the extended double bond conjugation associated with the experimentally reported enhanced inhibitory activity of F1 and MO10 against MAO-B.

Symmetric Zinbiel superalgebras

The notion of symmetric Zinbiel superalgebras is introduced. We prove that the nilpotency index of a symmetric Zinbiel superalgebra is not greater than 4 and describe two-generated symmetric Zinbiel algebras and odd generated superalgebras. We discuss identities of mono and binary symmetric Zinbiel and Leibniz algebras. It is proven that each quadratic Zinbiel algebra is 2-step nilpotent. Also, we study double extensions of symmetric Zinbiel algebras.

Odd-quadratic Lie superalgebras with a weak filiform module as an odd part

The aim of this work is to study a very special family of odd-quadratic Lie superalgebras g=g0¯⊕g1¯ such that g1¯ is a weak filiform g0¯-module (weak filiform type). We introduce this concept after having proved that the unique non-zero odd-quadratic Lie superalgebra (g,B) with g1¯ a filiform g0¯-module is the abelian 2-dimensional Lie superalgebra g=g0¯⊕g1¯ such that dimg0¯=dimg1¯=1. Let us note that in this context the role of the center of g is crucial. Thus, we obtain an inductive description of odd-quadratic Lie superalgebras of weak filiform type via generalized odd double extensions. Moreover, we obtain the classification, up to isomorphism, for the smallest possible dimensions, that is, six and eight.

Design of a 6 kV Beta-Ga2O3 PN Heterojunction Diode with Etched Double-Layered NiO with a Figure of Merit of 10 GW cm−2

This study proposes a NiO/β-Ga2O3 etched heterojunction extension termination design for NiO/β-Ga2O3 PN diode to fulfill the application of 6 kV high-voltage with a considerably outstanding power figure of merit 10 GW cm−2, validated and optimized by sentaurus TCAD software. The results indicated the optimum termination parameters L = 20 μm and DJET = 2.3 × 1013cm−2 for the single junction termination extension structure with 4000 V breakdown voltage, and L1 = L2 = 20 μm, DJET1 = 4 × 1013cm−2 and DJET2 = 2.3 × 1013 cm−2 for the double junction termination extension structure with 6000 V. Moreover, to evaluate the power figure of merit of the optimized devices, the specific resistances were extracted from the forward characterizations with setting appropriate electron and hole mobility from experiment results, which exhibited a power figure of merit of 4.7 GW cm−2 for the single junction termination extension and 10 GW cm−2 for double junction termination extension. Meanwhile, utilizing the Poisson equation calculated the ideal one-dimension electric field at NiO/Ga2O3 interface in Ga2O3 for the optimized devices, the single junction termination extension with 5.3 MV cm−1 at reverse 4000 V, and 7.3 MV cm−1 of the double junction termination extension incredibly approached Ga2O3 critical breakdown electric field of 8 MV cm−1 at reverse 6000 V, demonstrating our designed device structures possess immense potential for high-voltage power application.

Double Ore extensions of anti-angle type for Hopf algebras

<abstract><p>The aim of this article is to extend the structure of a bialgebra (Hopf algebra) which is connected graded as an algebra and generated in degree $ 1 $ to its double Ore extensions of anti-angle type. We construct two non-commutative and non-cocommutative Hopf algebras of infinite dimension and investigate the lifting of homological properties.</p></abstract>

Left invariant special Kähler structures

We construct left invariant special Kähler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted cartesian product of two special Kähler Lie algebras according to two linear representations by infinitesimal Kähler transformations. We also exhibit a double extension process of a special Kähler Lie algebra which allows us to get all simply connected special Kähler Lie groups with bi-invariant symplectic connections. All Lie groups constructed by performing this double extension process can be identified with a subgroup of symplectic (or Kähler) affine transformations of its Lie algebra containing a nontrivial 1-parameter subgroup formed by central translations. We show a characterization of left invariant flat special Kähler structures using étale Kähler affine representations, exhibit some immediate consequences of the constructions mentioned above, and give several non-trivial examples.

Flat left-invariant pseudo-Riemannian metrics on quadratic Lie groups

A flat quadratic Lie algebra (g,〈,〉,k) is a Lie algebra g endowed with a flat pseudo-Euclidean metric 〈,〉 and a quadratic structure k. In geometrical terms, it is a Lie algebra of a Lie group endowed with both a flat left-invariant pseudo-Riemannian metric and a bi-invariant metric. It is known that if a Lie algebra admits a symplectic form and a quadratic structure then it is a flat quadratic Lie algebra. In this paper, we show that there exist non-abelian Lie algebras which admit a flat Lorentzian metric and a quadratic structure. If g is such Lie algebra, then we show that g is a trivial central extension of R⋊H2k+1 and [g,g]≃H2k+1, where H2k+1 is the Heisenberg Lie algebra and k≥2. In particular, we prove that g is 3-step solvable, not nilpotent and hence admits no symplectic structure. We show that there exists only one non-abelian Lie algebra of dimension n≤6 which admits a flat Lorentzian metric and a quadratic structure. Using a variant of the double extension method, which conserves both the flat and the quadratic metrics, we prove that any Lie algebra which admits a flat Lorentzian metric and a quadratic structure, is a double extension of an abelian Lie algebra and we construct a large class of such (non-abelian) Lie algebras in higher dimensions. Oscillator Lie algebras denoted by gλ constitute an important class of quadratic Lie algebras which are of the form R⋊H2k+1. It is known that gλ admits no flat Lorentzian metric. We show in this paper that gλ admits a Ricci-flat Lorentzian metric. The paper also study the class of quadratic 2-step nilpotent Lie algebras (g,k) which constitute examples of flat quadratic Lie algebras since k is also flat in this case.

Towards understanding double extension twinning behaviors in magnesium alloy during uniaxial tension deformation

Multiple {101¯2} twin variants can be activated when the external tension loading is parallel to the normal direction of magnesium alloy plate. In the current work, we report that in addition to abundant primary {101¯2} twins, an atypical double twinning sequence, i.e. {101¯2}-{101¯2} double twin can also occur during uniaxial loading. The results of electron backscatter diffraction (EBSD) observation indicate that these secondary twins prefer to be formed at the regions where two primary {101¯2} twins belonging to different twin variant pairs interact with each other. Meanwhile, abundant dislocations activated can also be found within the primary twin containing secondary {101¯2} twin. Accordingly, the possible formation mechanisms of such secondary {101¯2} twin are proposed and discussed based on the local strain accommodation evaluated by a combination of geometrical compatibility parameter (m’) and Schmid factor. The analysis results suggest that the formation of these secondary twins may be closely related to the minimization of local strain incompatibility at the twinning boundary caused by the basal dislocation within the primary twins.

Higher Coverings of Racks and Quandles—Part II

This article is the second part of a series of three articles, in which we develop a higher covering theory of racks and quandles. This project is rooted in M. Eisermann’s work on quandle coverings, and the categorical perspective brought to the subject by V. Even, who characterizes coverings as those surjections which are categorically central, relatively to trivial quandles. We extend this work by applying the techniques from higher categorical Galois theory, in the sense of G. Janelidze, and in particular we identify meaningful higher-dimensional centrality conditions defining our higher coverings of racks and quandles. In this second article (Part II), we show that categorical Galois theory applies to the inclusion of the category of coverings into the category of surjective morphisms of racks or of quandles. We characterise the induced Galois-theoretic concepts of trivial covering, normal covering and covering in this two-dimensional context. The latter is described via our definition and study of double coverings, also called algebraically central double extensions of racks (and quandles). We define a suitable and well-behaved commutator which captures the zero-, one- and two-dimensional concepts of centralization in the category of quandles. We keep track of the links with the corresponding concepts in the category of groups.

Algorithm for Generating Extra-Long Gordon–Mills–Welch Sequences

The modification of an algorithm for determining polynomial multipliers hci(x) of test polynomial hGMWS(x), which is the main component of the method for synthesizing Gordon–Mills–Welch sequences (GMWSs), is set as a basis for developing the software implementation of the algorithm for generating extra-long GMWSs having a two-level periodic autocorrelation function and high structural latency and generated over the finite field with double extension GF(2S) = GF[(2m)n]. In the modified algorithm, the expressions for the number of operations in computing the vector of alternatives, from which the vector of decimation indices is formed, are derived for different values of parameter n. These expressions also serve as upper estimates for the number of summed sequences. To generate extra-long GMWSs with periods from N = 212 – 1 = 4095 to N = 220 – 1 = 1 048 575, the sets of vectors of decimation indices are obtained for admissible values of parameters m and n.

Hom Leibniz Superalgebras With Supersymmetric Invariant Nondegenerate Bilinear Forms

In this paper we study Hom Leibniz superalgebras endowed with a supersymmetric, nondegenerate and invariant bilinear form. Such Hom Leibniz superalgebras are called quadratic Hom Leibniz superalgebras. We show that every quadratic Hom Leibniz superalgebra is symmetric. After introducing representation of Hom Leibniz superalgebras, we prove that Hom Leibniz superalgebras are quadratic if and only if the adjoint representation and its dual representation are equivalent. We also extend the notion of T*-extension to Hom Leibniz superalgebras. Finally, by using double extension, we describe inductively quadratic regular Hom Leibniz superalgebras.

Double Extensions on Riemannian Ricci Nilsolitons

In this paper, we study left invariant Lorentz algebraic Ricci solitons on nilpotent Lie groups. Based on the concept of Lorentz datum and the technique of double extensions, we are able to construct Lorentz Ricci nilsolitons from any Riemannian Ricci nilsoliton. Conversely, we prove that any Lorentz Ricci nilsoliton with degenerate center is a double extension of a Riemannian Ricci nilsoliton with respect to a Lorentz data. Moreover, we provide a strategy to classify Lorentz Ricci nilsolitons with degenerate center. As an application, we present a large number of Lorentz Ricci nilsolitons based on Abelian Riemannian Ricci solitons. Finally, a family of left invariant Lorentz Ricci nilsolitons on a 9-dimensional nilpotent Lie algebra is given.