On the Ricci curvature of solvable metric lie algebras with two-step nilpotent derived algebras

SYNOPSISAlgebras which are nilpotent and anti-commutative are studied. Canonical forms are found for all such algebras of dimension n whose centres have dimension n−r (r < 3), and characters are given which enable any two non-isomorphic algebras to be distinguished.A metrisable Lie algebra is a Lie algebra for which there is a non-singular, symmetric, adjoint-invariant bilinear form a(λ, μ), and such an algebra is reduced if its centre is contained in its derived algebra. The importance of the reduced algebras follows from the fact that every metrisable Lie algebra is the direct sum of a reduced metrisable Lie algebra and an abelian Lie algebra. Tsou (Thesis 1955) introduced metrisable Lie algebras, and obtained canonical forms for all real reduced metrisable Lie algebras whose derived algebras have dimension 3. We conclude this paper by providing an alternative derivation, two of the algebras being nilpotent.

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On the Ricci curvature of nonunimodular solvable metric Lie algebras of small dimension

Siberian Advances in Mathematics ◽

10.3103/s1055134411020015 ◽

2011 ◽

Vol 21 (2) ◽

pp. 81-99

Author(s):

M. S. Chebarykov

Keyword(s):

Lie Algebras ◽

Ricci Curvature ◽

Small Dimension

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On the Ricci curvature of three-dimensional metric lie algebras

Владикавказский математический журнал ◽

10.23671/vnc.2014.1.7424 ◽

2014 ◽

Author(s):

М.С. Чебарыков

Keyword(s):

Lie Algebras ◽

Ricci Curvature ◽

Three Dimensional

В работе получена классификация возможных сигнатур кривизны Риччи левоинвариантных римановых метрик на трехмерных группах Ли, являющаяся уточнением некоторых результатов Дж. Милнора. В качестве вспомогательного результата получена классификация трехмерных неунимодулярных метрических алгебр Ли.

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Lie Groups, Lie Algebras, Cohomology and some Applications in Physics

10.1017/cbo9780511599897 ◽

1995 ◽

Cited By ~ 90

Author(s):

Josi A. de Azcárraga ◽

Josi M. Izquierdo

Keyword(s):

Lie Groups ◽

Lie Algebras

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The rigidity of some solvable Lie algebras

Uzbek Mathematical Journal ◽

10.29229/uzmj.2018-2-4 ◽

2018 ◽

Vol 2018 (2) ◽

pp. 43-49

Author(s):

R.K. Gaybullaev ◽

Kh.A. Khalkulova ◽

J.Q. Adashev

Keyword(s):

Lie Algebras ◽

Solvable Lie Algebras

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Integrable two-dimensional dynamical systems and the characteristic Lie algebras

Proceedings of the Mavlyutov Institute of Mechanics ◽

10.21662/uim2007.1.023 ◽

2007 ◽

Vol 5 ◽

pp. 195-200

Author(s):

A.V. Zhiber ◽

O.S. Kostrigina

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Lie Algebra ◽

Lie Algebras ◽

Second Order ◽

System Of Equations ◽

Two Dimensional ◽

Finite Dimensional ◽

Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

Multiphase Systems ◽

10.21662/mfs2018.3.009 ◽

2018 ◽

Vol 13 (3) ◽

pp. 59-63 ◽

Cited By ~ 1

Author(s):

D.T. Siraeva

Keyword(s):

Equation Of State ◽

Lie Algebra ◽

Lie Algebras ◽

Gas Dynamics ◽

Hydrodynamic Type ◽

System Of Equations ◽

Two Dimensional ◽

Equations Of Gas Dynamics ◽

Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

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