Classification and Identification of Lie
                    Algebras

Šnobl, Libor; Winternitz, P.

doi:10.1090/crmm/033

Cited by 105 publications

(243 citation statements)

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“…There are eleven types of three-dimensional real Lie algebras; in fact, nine algebras and two parametrized infinite families of algebras (see, e.g., [18], [22], [26]). In terms of an (appropriate) ordered basis (E 1 , E 2 , E 3 ), the commutation operation is given by…”

Section: A Three-dimensional Lie Algebrasmentioning

confidence: 99%

Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

Biggs

2017

Communications in Mathematics

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Abstract. We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine the isometry group for each normalized structure and hence characterize for exactly which structures (and groups) the isotropy subgroup of the identity is contained in the group of automorphisms of the Lie group. It turns out (in both the Riemannian and sub-Riemannian cases) that for most structures any isometry is the composition of a left translation and a Lie group automorphism.

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Section: A Three-dimensional Lie Algebrasmentioning

confidence: 99%

Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

Biggs

2017

Communications in Mathematics

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“…For general q the algebra is A 2 ⊕ A 2 which can also be written as 2A 2 . (We make use of the Mubarakzyanov Classification Scheme [5][6][7][8] (see also [9,11,13]) throughout this paper.) In the case of q = 1 we have the well-known Kummer-Schwarz Equation with the algebra 2sl(2, R) or 2A 3,8 .…”

Section: Symmetry Propertiesmentioning

confidence: 99%

Further Generalisations of the Kummer-Schwarz Equation: Algebraic and Singularity Properties

2017

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Abstract.The Kummer-Schwarz Equation, 2y y − 3(y ) 2 = 0, has a generalisation, (n − 1)y (n−2) y (n) − ny (n−1) 2 = 0, which shares many properties with the parent form (see Sinuvasan R, Tamizhmani K M & Leach P G L, Algebraic and singularity properties of a class of generalisations of the Kummer-Schwarz equation differ, Equ Dyn Syst (2016), doi:10.1007/s12591-016-0327-5) in terms of symmetry and singularity. All equations of the class are integrable in closed form. Here we introduce a new class, (n+q −2)y (n−2) y (n) − (n+q −1)y (n−1) 2 = 0, which has different integrability and singularity properties.

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“…The simplest case of a non-trivial Vessiot-Guldberg-Lie algebra is given by a Lie algebra in dimension two, as it is either Abelian or isomorphic to the affine algebra a 2 on the plane [15] (in the following, the reader is led to this reference for the elementary properties of Lie algebras; for classifications in low dimensions, see [15,16]). Suppose thus that the system Equation (8) admits a Vessiot-Guldberg-Lie algebra L VG of dimension two (it is implicitly assumed that it does not admit a smaller dimensional Lie algebra L VG ).…”

Section: Vessiot-guldberg-lie Algebras With R ≤ 3 For Scalar Sode Sysmentioning

confidence: 99%

“…At this stage, the problem corresponds essentially to classifying three-dimensional Lie algebras, a well-known problem solved in any standard reference [15]. We merely indicate that it suffices to consider the following three cases:…”

Section: Vessiot-guldberg-lie Algebras With R ≤ 3 For Scalar Sode Sysmentioning

confidence: 99%

“…A 3,9 . Once the possible Lie algebras g have been deduced, the precise functions in the realization Equation (15), as well as the differential equations admitting g as Vessiot-Guldberg-Lie algebra L VG are obtained by solving the successive conditions imposed by the commutators of g. As an illustration of the procedure, we give the details for the Lie algebras r 1,ε , the remaining cases being obtained in a completely analogous way.…”

Section: Vessiot-guldberg-lie Algebras With R ≤ 3 For Scalar Sode Sysmentioning

confidence: 99%

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Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

Campoamor-Stursberg

2016

Symmetry

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Abstract:A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebra. In particular, well-known types, such as the Milne-Pinney or Kummer-Schwarz equations, are recovered as special cases of this classification. The analogous problem for systems of second-order differential equations in the real plane is considered for a special case that enlarges the generalized Ermakov systems.

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Classification and Identification of Lie Algebras

Cited by 105 publications

References 0 publications

Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

Further Generalisations of the Kummer-Schwarz Equation: Algebraic and Singularity Properties

Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

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