Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras

Lucas, J. de; Wysocki, D.

doi:10.3390/sym13030465

Cited by 3 publications

(3 citation statements)

References 40 publications

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“…In general, for solvable and nilpotent Lie algebras many of their Lie bialgebra structures are noncoboundaries; in this respect, see [50,51] and references therein for the classifcation of 3D Lie bialgebras. Finally, we point out that, very recently, the classification of 4D indecomposable coboundary Lie bialgebras has been carried out in [52], which shows how the difficulties of this task grow when the dimensions of the Lie bialgebras increase.…”

Section: Lie Bialgebras and Quantum Algebrasmentioning

confidence: 85%

Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications

Gutierrez-Sagredo,

Herranz

2021

Preprint

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The Cayley-Klein (CK) formalism is applied to the real algebra so(5) by making use of four graded contraction parameters describing in a unified setting 81 Lie algebras, which cover the (anti-)de Sitter, Poincaré, Newtonian and Carrollian algebras. Starting with the Drinfel'd-Jimbo real Lie bialgebra for so(5) together with its Drinfel'd double structure, we obtain the corresponding CK bialgebra and the CK r-matrix coming from a Drinfel'd double. As a novelty, we construct the (first-order) noncommutative CK spaces of points, lines, 2-planes and 3-hyperplanes, studying their structural properties. By requiring to deal with real structures, it comes out that there exist 63 specific real Lie bialgebras together with their sets of four noncommutative spaces. Furthermore, we find 14 classical r-matrices coming from Drinfel'd doubles, obtaining new results for the de Sitter so(4, 1) and anti-de Sitter so(3, 2) and for some of their contractions. These geometric results are exhaustively applied onto the (3+1)D kinematical algebras, not only considering the usual (3+1)D spacetime but also the 6D space of lines. We establish different assignations between the geometrical CK generators and the kinematical ones which convey physical identifications for the CK contraction parameters in terms of the cosmological constant/curvature Λ and speed of light c. We finally obtain four classes of kinematical r-matrices together with their noncommutative spacetimes and spaces of lines, comprising all κ-deformations as particular cases.

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Section: Lie Bialgebras and Quantum Algebrasmentioning

confidence: 85%

Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications

Gutierrez-Sagredo,

Herranz

2021

Preprint

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“…Moreover, every real non-abelian three-dimensional Lie algebra is isomorphic to (E, [•, •]), where E is a three-dimensional vector space and the Lie bracket is given on a canonical basis {e 1 , e 2 , e 3 } of E by one of the cases in table 1 (see e.g. [36,38]).…”

Section: Existence Of Invariant Contact Forms For Lie Systemsmentioning

confidence: 99%

Contact Lie systems: theory and applications

Lucas

Rivas

2023

J. Phys. A: Math. Theor.

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A Lie system is a time-dependent system of differential equations describing the integral curves of a time-dependent vector field that can be considered as a curve in a finite-dimensional Lie algebra of vector fields $V$. We call $V$ a Vessiot--Guldberg Lie algebra.We define and analyse contact Lie systems, namely Lie systems admitting a Vessiot--Guldberg Lie algebra of Hamiltonian vector fields relative to a contact manifold. We also study contact Lie systems of Liouville type, which are invariant relative to the flow of a Reeb vector field. Liouville theorems, contact Marsden--Weinstein reductions, and Gromov non-squeezing theorems are developed and applied to contact Lie systems. Contact Lie systems on three-dimensional Lie groups with Vessiot--Guldberg Lie algebras of right-invariant vector fields and associated with left-invariant contact forms are classified. Our results are illustrated by examples with relevant physical and mathematical applications, e.g. Schwarz equations, Brockett systems, quantum mechanical systems, etc. Finally, a Poisson coalgebra method to derive superposition rules for contact Lie systems of Liouville is developed.

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“…In general, for solvable and nilpotent Lie algebras, many of their Lie bialgebra structures are non-coboundaries; in this respect, see [54,55] and the references therein for the classification of 3D Lie bialgebras. Finally, very recently, the classification of 4D inde-composable coboundary Lie bialgebras was carried out in [56], which showed how the difficulties of this task grow when the dimensions of the Lie bialgebras increase.…”

Section: Lie Bialgebras and Quantum Algebrasmentioning

confidence: 99%

Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

Gutiérrez-Sagredo

Herranz

2021

Symmetry

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The Cayley–Klein (CK) formalism is applied to the real algebra so(5) by making use of four graded contraction parameters describing, in a unified setting, 81 Lie algebras, which cover the (anti-)de Sitter, Poincaré, Newtonian and Carrollian algebras. Starting with the Drinfel’d–Jimbo real Lie bialgebra for so(5) together with its Drinfel’d double structure, we obtain the corresponding CK bialgebra and the CK r-matrix coming from a Drinfel’d double. As a novelty, we construct the (first-order) noncommutative CK spaces of points, lines, 2-planes and 3-hyperplanes, studying their structural properties. By requiring dealing with real structures, we found that there exist 63 specific real Lie bialgebras together with their sets of four noncommutative spaces. Furthermore, we found 14 classical r-matrices coming from Drinfel’d doubles, obtaining new results for the de Sitter so(4,1) and anti-de Sitter so(3,2) as well as for some of their contractions. These geometric results were exhaustively applied onto the (3 + 1)D kinematical algebras, considering not only the usual (3 + 1)D spacetime but also the 6D space of lines. We established different assignations between the geometrical CK generators and the kinematical ones, which convey physical identifications for the CK contraction parameters in terms of the cosmological constant/curvature Λ and the speed of light c. We, finally, obtained four classes of kinematical r-matrices together with their noncommutative spacetimes and spaces of lines, comprising all κ-deformations as particular cases.

show abstract

Darboux Families and the Classification of Real Four-Dimensional Indecomposable Coboundary Lie Bialgebras

Cited by 3 publications

References 40 publications

Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications

Cayley-Klein Lie bialgebras: Noncommutative spaces, Drinfel'd doubles and kinematical applications

Contact Lie systems: theory and applications

Cayley–Klein Lie Bialgebras: Noncommutative Spaces, Drinfel’d Doubles and Kinematical Applications

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