Classification of real three-dimensional Lie bialgebras and their Poisson-Lie groups

Abstract

Classical r-matrices of the three-dimensional real Lie bialgebras are obtained. In this way all three-dimensional real coboundary Lie bialgebras and their types (triangular, quasitriangular or factorizable) are classified. Then, by using the Sklyanin bracket, the Poisson structures on the related Poisson-Lie groups are obtained.

17 pages

References in corpus (12)

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• Poisson-Lie T-duality
• Poisson-Lie T-plurality
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Cited by in corpus (17)

• Classification of two and three dimensional Lie super-bialgebras
• Classification of four and six dimensional Drinfel'd superdoubles
• The gl(1|1) Lie superbialgebras
• Yang-Baxter deformations of WZW model on the Heisenberg Lie group
• Classical r-matrices of two and three dimensional Lie super-bialgebras and their Poisson-Lie supergroups
• Quantizations of D=3 Lorentz symmetry
• Classification of real low dimensional Jacobi(generalized)-Lie bialgebras
• Jacobi structures on real two- and three-dimensional Lie groups and their Jacobi-Lie systems
• Classical r-matrices of real low-dimensional Jacobi-Lie bialgebras and their Jacobi-Lie groups
• Jacobi-Lie Hamiltonian systems on real low-dimensional Jacobi-Lie groups and their Lie symmetries
• Lie superbialgebra structures on the Lie superalgebra $({\mathcal{C}}^3+\mathcal{A})$ and deformation of related integrable Hamiltonian systems
• Classification of four dimensional real Lie bialgebras of symplectic type and their Poisson-Lie groups
• T-dualization of Gödel string cosmologies via Poisson-Lie T-duality approach
• Classification of real three-dimensional Poisson-Lie groups
• M2 to D2 and vice versa by 3-Lie and Lie bialgebra
• Non-Abelian T-duality of $Ad{S}_{d\le 3}$ families by Poisson-Lie T-duality
• Integrable bi-Hamiltonian systems by Jacobi structure on real three-dimensional Lie groups