Classical r-matrices via semidualisation
arXiv:1307.6485 · doi:10.1063/1.4824704
Abstract
We study the interplay between double cross sum decompositions of a given Lie algebra and classical r-matrices for its semidual. For a class of Lie algebras which can be obtained by a process of generalised complexification we derive an expression for classical r-matrices of the semidual Lie bialgebra in terms of the data which determines the decomposition of the original Lie algebra. Applied to the local isometry Lie algebras arising in three-dimensional gravity, decomposition and semidualisation yields the main class of non-trivial r-matrices for the Euclidean and Poincare group in three dimensions. In addition, the construction links the r-matrices with the Bianchi classification of three dimensional real Lie algebras.
21 pages, 1 figure, typos corrected
References in corpus (5)
- Generalised Chern-Simons actions for 3d gravity and kappa-Poincare symmetry
- Quaternionic and Poisson-Lie structures in 3d gravity: the cosmological constant as deformation parameter
- Geometrical (2+1)-gravity and the Chern-Simons formulation: Grafting, Dehn twists, Wilson loop observables and the cosmological constant
- On the semiduals of local isometry groups in 3d gravity
- 6J Symbols Duality Relations