scholarly journals Extensions of hom-Lie color algebras

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Abdoreza Armakan ◽  
Sergei Silvestrov ◽  
Mohammad Reza Farhangdoost
Keyword(s):  
Differential Geometry ◽  
Bianchi Identity ◽  
Geometrical Interpretation ◽  
Abelian Extensions ◽  
The Third ◽  
Lie Color Algebras

Abstract In this paper, we study (non-Abelian) extensions of a given hom-Lie color algebra and provide a geometrical interpretation of extensions. In particular, we characterize an extension of a hom-Lie color algebra {\mathfrak{g}} by another hom-Lie color algebra {\mathfrak{h}} and discuss the case where {\mathfrak{h}} has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, curvature and the Bianchi identity for possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie color algebras, i.e., we show that in order to have an extendible hom-Lie color algebra, there should exist a trivial member of the third cohomology.

Geometric aspects of extensions of hom-Lie superalgebras

2017 ◽  
Vol 14 (06) ◽  
pp. 1750085 ◽  
Author(s):  
A. R. Armakan ◽  
M. R. Farhangdoost
Keyword(s):  
Differential Geometry ◽  
Bianchi Identity ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Abelian Extensions

In this paper, we deal with (non-abelian) extensions of a given hom-Lie superalgebra and find a cohomological obstacle to the existence of extensions of hom-Lie superalgebras. Moreover, the setting of covariant exterior derivatives, super connection, curvature and the Bianchi identity in differential geometry has been studied.

Curvature Expressions for the Large Displacement Analysis of Planar Beam Motions

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
Yinhuan Zheng ◽  
Ahmed A. Shabana ◽  
Dayu Zhang
Keyword(s):  
Differential Geometry ◽  
Finite Element ◽  
Beam Element ◽  
Elastic Force ◽  
Large Displacement ◽  
Displacement Analysis ◽  
The Third ◽  
Shear Deformable Beam ◽  
Approach Method ◽  
Shear Deformable

While several curvature expressions have been used in the literature, some of these expressions differ from basic geometry definitions and lead to kinematic coupling between bending and shear deformations. This paper uses three different elastic force formulations in order to examine the effect of the curvature definition in the large displacement analysis of beams. In the first elastic force formulation, a general continuum mechanics approach (method 1) based on the nonlinear strain–displacement relationship is used. The second approach (method 2) is based on a classical nonlinear beam theory, in which a curvature expression consistent with differential geometry and independent of the shear deformation is used. The third elastic force formulation (method 3) employs a curvature expression that depends on the shear angle. In order to examine numerically the effect of using different curvature definitions, three different planar beam elements are used. The first element (element I) is the fully parameterized absolute nodal coordinate formulation (ANCF) shear deformable beam element. The second element (element II) is an ANCF consistent rotation-based formulation (CRBF) shear deformable beam element obtained from element I by consistently replacing the position gradient vectors by rotation parameters. The third element (element III) is a low-order bilinear ANCF/CRBF finite element in which nonzero differential geometry-based curvature definition cannot be obtained because of the low order of interpolation. Numerical results are obtained using the three elastic force formulations and the three finite elements in order to shed light on the definition of bending and shear in the large displacement analysis of beams. The results obtained in this investigation show that the use of method 2, with a penalty formulation that restricts the excessive cross section deformation, can improve significantly the convergence of the ANCF finite element.

scholarly journals $T^*$-extensions and abelian extensions of hom-Lie color algebras

2017 ◽  
pp. 123-142
Author(s):  
Bing Sun ◽  
Liangyun Chen ◽  
Yan Liu
Keyword(s):  
Abelian Extension ◽  
Closed Field ◽  
Infinitesimal Deformation ◽  
Algebraically Closed Field ◽  
Abelian Extensions ◽  
Finite Dimensional ◽  
Lie Color Algebras

We study hom-Nijenhuis operators, $T^ \ast$-extensions and abelian extensions of hom-Lie color algebras. We show that the infinitesimal deformation generated by a hom-Nijenhuis operator is trivial. Many properties of a hom-Lie color algebra can be lifted to its $T^ \ast$-extensions such as nilpotency, solvability and decomposition. It is proved that every finite-dimensional nilpotent quadratic hom-Lie color algebra over an algebraically closed field of characteristic not 2 is isometric to a $T^ \ast$-extension of a nilpotent Lie color algebra. Moreover, we introduce abelian extensions of hom-Lie color algebras and show that there is a representation and a 2-cocycle, associated to any abelian extension.

scholarly journals Surfaces and Curves Induced by Nonlinear Schrödinger-Type Equations and Their Spin Systems

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1827
Author(s):  
Akbota Myrzakul ◽  
Gulgassyl Nugmanova ◽  
Nurzhan Serikbayev ◽  
Ratbay Myrzakulov
Keyword(s):  
Differential Geometry ◽  
Spin Systems ◽  
Heisenberg Ferromagnet ◽  
Integrable Equations ◽  
Nonlinear Schrödinger ◽  
Space Curves ◽  
The Third ◽  
Integrable Spin ◽  
Frenet Equation ◽  
Basis Vectors

In recent years, symmetry in abstract partial differential equations has found wide application in the field of nonlinear integrable equations. The symmetries of the corresponding transformation groups for such equations make it possible to significantly simplify the procedure for establishing equivalence between nonlinear integrable equations from different areas of physics, which in turn open up opportunities to easily find their solutions. In this paper, we study the symmetry between differential geometry of surfaces/curves and some integrable generalized spin systems. In particular, we investigate the gauge and geometrical equivalence between the local/nonlocal nonlinear Schrödinger type equations (NLSE) and the extended continuous Heisenberg ferromagnet equation (HFE) to investigate how nonlocality properties of one system are inherited by the other. First, we consider the space curves induced by the nonlinear Schrödinger-type equations and its equivalent spin systems. Such space curves are governed by the Serret–Frenet equation (SFE) for three basis vectors. We also show that the equation for the third of the basis vectors coincides with the well-known integrable HFE and its generalization. Two other equations for the remaining two vectors give new integrable spin systems. Finally, we investigated the relation between the differential geometry of surfaces and integrable spin systems for the three basis vectors.

scholarly journals Gray identities, canonical connection and integrability

2010 ◽  
Vol 53 (3) ◽  
pp. 657-674 ◽  
Author(s):  
Antonio J. Di Scala ◽  
Luigi Vezzoni
Keyword(s):  
Curvature Tensor ◽  
Bianchi Identity ◽  
Main Tool ◽  
Riemann Curvature Tensor ◽  
The Third ◽  
Hermitian Connection ◽  
Almost Kähler ◽  
Almost Hermitian Structures ◽  
Kähler Structures ◽  
Hermitian Structures

AbstractWe characterize quasi-Kähler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related to the third Gray identity and in the almost-Kähler case implies the integrability. Our main tool is the existence of generalized holomorphic frames previously introduced by the second author. By using such frames we also give a simpler and shorter proof of a theorem of Goldberg. Furthermore, we study almost-Hermitian structures having the curvature tensor associated to the canonical Hermitian connection equal to zero. We show some explicit examples of quasi-Kähler structures on the Iwasawa manifold having the Hermitian curvature vanishing and the Riemann curvature tensor satisfying the second Gray identity.

scholarly journals Riccati PDEs That Imply Curvature-Flatness

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 537
Author(s):  
Iulia Hirica ◽  
Constantin Udriste ◽  
Gabriel Pripoae ◽  
Ionel Tevy
Keyword(s):  
Differential Geometry ◽  
Partial Differential Equations ◽  
Projection Operator ◽  
Point Of View ◽  
Vector Spaces ◽  
Partial Differential ◽  
The Third ◽  
Kronecker Delta ◽  
Symmetric Connection ◽  
Differential Relations

In this paper the following three goals are addressed. The first goal is to study some strong partial differential equations (PDEs) that imply curvature-flatness, in the cases of both symmetric and non-symmetric connection. Although the curvature-flatness idea is classic for symmetric connection, our main theorems about flatness solutions are completely new, leaving for a while the point of view of differential geometry and entering that of PDEs. The second goal is to introduce and study some strong partial differential relations associated to curvature-flatness. The third goal is to introduce and analyze some vector spaces of exotic objects that change the meaning of a generalized Kronecker delta projection operator, in order to discover new PDEs implying curvature-flatness. Significant examples clarify some ideas.

scholarly journals Nonassociative Algebras: A Framework for Differential Geometry

2003 ◽  
Vol 2003 (60) ◽  
pp. 3777-3795 ◽  
Author(s):  
Lucian M. Ionescu
Keyword(s):  
Differential Geometry ◽  
Lie Algebra ◽  
Vector Fields ◽  
Bianchi Identity ◽  
Affine Connection ◽  
Natural Generalization ◽  
Nonassociative Algebra ◽  
Lie Bracket ◽  
Algebra Of Functions ◽  
Starting Point

A nonassociative algebra endowed with a Lie bracket, called atorsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted as a connection. This provides a framework for differential geometry on a formal manifold with a formal connection. A torsion algebra is a natural generalization of pre-Lie algebras which appear as the “torsionless” case. The starting point is the observation that the associator of a nonassociative algebra is essentially the curvature of the corresponding Hochschild quasicomplex. It is a cocycle, and the corresponding equation is interpreted as Bianchi identity. The curvature-associator-monoidal structure relationships are discussed. Conditions on torsion algebras allowing to construct an algebra of functions, whose algebra of derivations is the initial Lie algebra, are considered. The main example of a torsion algebra is provided by the pre-Lie algebra of Hochschild cochains of aK-module, with Lie bracket induced by Gerstenhaber composition.

scholarly journals GEOMETRIC AND EXTENSOR ALGEBRAS AND THE DIFFERENTIAL GEOMETRY OF ARBITRARY MANIFOLDS

2007 ◽  
Vol 04 (07) ◽  
pp. 1117-1158 ◽  
Author(s):  
V. V. FERNÁNDEZ ◽  
A. M. MOYA ◽  
W. A. RODRIGUES
Keyword(s):  
Differential Geometry ◽  
Moving Frame ◽  
Geometrical Structures ◽  
Metric Tensor ◽  
The Third ◽  
Covariant Derivatives ◽  
Open Set ◽  
Structure Equations ◽  
Parallelism Structure ◽  
Derivatives Of

We give in this paper which is the third in a series of four a theory of covariant derivatives of representatives of multivector and extensor fields on an arbitrary open set U ⊂ M, based on the geometric and extensor calculus on an arbitrary smooth manifold M. This is done by introducing the notion of a connection extensor field γ defining a parallelism structure on U ⊂ M, which represents in a well-defined way the action on U of the restriction there of some given connection ∇ defined on M. Also we give a novel and intrinsic presentation (i.e. one that does not depend on a chosen orthonormal moving frame) of the torsion and curvature fields of Cartan's theory. Two kinds of Cartan's connection operator fields are identified, and both appear in the intrinsic Cartan's structure equations satisfied by the Cartan's torsion and curvature extensor fields. We introduce moreover a metrical extensor g in U corresponding to the restriction there of given metric tensor g defined on M and also introduce the concept of a geometric structure(U, γ ,g) for U ⊂ M and study metric compatibility of covariant derivatives induced by the connection extensor γ. This permits the presentation of the concept of gauge (deformed) derivatives which satisfy noticeable properties useful in differential geometry and geometrical theories of the gravitational field. Several derivatives of operators in metric and geometrical structures, like ordinary and covariant Hodge co-derivatives and some duality identities are exhibited.

Progress report on 21-cm observations at low latitude between longitudes (l 11) 0° and 66°

1967 ◽  
Vol 31 ◽  
pp. 177-179
Author(s):  
W. W. Shane
Keyword(s):  
Preliminary Report ◽  
Progress Report ◽  
Galactic Centre ◽  
Low Latitude ◽  
The Third ◽  
Introductory Lecture ◽  
Centre Region

In the course of several 21-cm observing programmes being carried out by the Leiden Observatory with the 25-meter telescope at Dwingeloo, a fairly complete, though inhomogeneous, survey of the regionl11= 0° to 66° at low galactic latitudes is becoming available. The essential data on this survey are presented in Table 1. Oort (1967) has given a preliminary report on the first and third investigations. The third is discussed briefly by Kerr in his introductory lecture on the galactic centre region (Paper 42). Burton (1966) has published provisional results of the fifth investigation, and I have discussed the sixth in Paper 19. All of the observations listed in the table have been completed, but we plan to extend investigation 3 to a much finer grid of positions.

Sign in / Sign up

Export Citation Format

Share Document