scholarly journals Classification of real three-dimensional Poisson–Lie groups

2012 ◽  
Vol 45 (17) ◽  
pp. 175204 ◽  
Author(s):  
Ángel Ballesteros ◽  
Alfonso Blasco ◽  
Fabio Musso
Keyword(s):  
Lie Groups ◽  
Three Dimensional ◽  
Poisson Lie Groups

Classification of 3-dimensional left-invariant almost paracontact metric structures

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Giovanni Calvaruso ◽  
Antonella Perrone
Keyword(s):  
Lie Groups ◽  
Three Dimensional ◽  
Complete Classification ◽  
Natural Assumption ◽  
Geometric Properties ◽  
3 Dimensional ◽  
Metric Structures

AbstractWe study left-invariant almost paracontact metric structures on arbitrary three-dimensional Lorentzian Lie groups. We obtain a complete classification and description under a natural assumption, which includes relevant classes as normal and almost para-cosymplectic structures, and we investigate geometric properties of these structures.

scholarly journals Classification of four-dimensional real Lie bialgebras of symplectic type and their Poisson–Lie groups

2017 ◽  
Vol 190 (1) ◽  
pp. 1-17 ◽  
Author(s):  
J. Abedi-Fardad ◽  
A. Rezaei-Aghdam ◽  
Gh. Haghighatdoost
Keyword(s):  
Lie Groups ◽  
Lie Bialgebras ◽  
Poisson Lie Groups

scholarly journals On Invariant Semisymmetric Connections on Three-Dimensional Non-Unimodular Lie Groups with the Metric of the Ricci Soliton

2021 ◽  
pp. 86-90
Author(s):  
D.V. Vylegzhanin ◽  
P.N. Klepikov ◽  
E.D. Rodionov ◽  
O.P. Khromova
Keyword(s):  
Lie Groups ◽  
Einstein Metrics ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Natural Generalization ◽  
Ricci Solitons ◽  
Tensor Fields ◽  
Left Invariant Riemannian Metric ◽  
Riemannian Case

Metric connections with vector torsion, or semisymmetric connections, were first discovered by E. Cartan. They are a natural generalization of the Levi-Civita connection. The properties of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano, and other mathematicians. Ricci solitons are the solution to the Ricci flow and a natural generalization of Einstein's metrics. In the general case, they were investigated by many mathematicians, which was reflected in the reviews by H.-D. Cao, R.M. Aroyo — R. Lafuente. This question is best studied in the case of trivial Ricci solitons, or Einstein metrics, as well as the homogeneous Riemannian case. This paper investigates semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional non-unimodularLie groups with the left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that there are nontrivial invariant semisymmetric connections in this case. In addition, it is shown that there are nontrivial invariant Ricci solitons.

scholarly journals Mathematical modeling in the study of semisymmetric connections on three-dimensional Lie groups with the metric of the Ricci soliton

2021 ◽  
Vol 60 (1) ◽  
pp. 23-29
Author(s):  
Pavel N. Klepikov ◽  
Evgeny D. Rodionov ◽  
Olesya P. Khromova
Keyword(s):  
Lie Groups ◽  
Einstein Metrics ◽  
Three Dimensional ◽  
Ricci Soliton ◽  
Riemannian Metric ◽  
Natural Generalization ◽  
Tensor Fields ◽  
Parallel Transfer ◽  
Left Invariant Riemannian Metric

Semisymmetric connections were first discovered by E. Cartan and are a natural generalization of the Levi-Civita connection. The properties of the parallel transfer of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano and other mathematicians. In this paper, a mathematical model is constructed for studying semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional unimodular Lie groups with left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that in this case there are nontrivial invariant semisimetric connections. Previously, the authors carried out similar studies in the class of Einstein metrics.

Cheng–Shen conjecture in Finsler geometry

2020 ◽  
Vol 31 (04) ◽  
pp. 2050030
Author(s):  
M. Atashafrouz ◽  
B. Najafi
Keyword(s):  
Lie Groups ◽  
Finsler Geometry ◽  
Three Dimensional ◽  
Closed Manifold ◽  
Classification Theorem ◽  
Randers Metric ◽  
Counter Example ◽  
Randers Metrics

The well-known Cheng–Shen conjecture says that every [Formula: see text]-quadratic Randers metric on a closed manifold is a Berwald metric. The class of [Formula: see text]-quadratic Randers metrics contains the class of generalized Douglas–Weyl Randers metrics. In this paper, we give a classification of left-invariant Randers metrics of generalized Douglas–Weyl type on three-dimensional Lie groups. Based on our classification theorem, we find a counter-example for the Cheng–Shen conjecture.

scholarly journals Topological Loops with Decomposable Solvable Multiplication Groups

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Ameer Al-Abayechi ◽  
Ágota Figula
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Three Dimensional ◽  
Solvable Lie Groups ◽  
3 Dimensional ◽  
Multiplication Group ◽  
Simply Connected ◽  
Solvable Lie Group

AbstractIn this paper we deal with the class $$\mathcal {C}$$ C of decomposable solvable Lie groups having dimension six. We determine those Lie groups in $$\mathcal {C}$$ C and their subgroups which are the multiplication groups Mult(L) and the inner mapping groups Inn(L) for three-dimensional connected simply connected topological loops L. This result completes the classification of the at most 6-dimensional solvable multiplication Lie groups of the loops L. Moreover, we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two.

scholarly journals Classification of real low-dimensional Jacobi (generalized)–Lie bialgebras

2016 ◽  
Vol 14 (01) ◽  
pp. 1750007 ◽  
Author(s):  
A. Rezaei-Aghdam ◽  
M. Sephid
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Lie Bialgebras ◽  
Structure Constants ◽  
Definition Of ◽  
Low Dimensional

We describe the definition of Jacobi (generalized)–Lie bialgebras [Formula: see text] in terms of structure constants of the Lie algebras [Formula: see text] and [Formula: see text] and components of their 1-cocycles [Formula: see text] and [Formula: see text] in the basis of the Lie algebras. Then, using adjoint representations and automorphism Lie groups of Lie algebras, we give a method for classification of real low-dimensional Jacobi–Lie bialgebras. In this way, we obtain and classify real two- and three-dimensional Jacobi–Lie bialgebras.

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