2013/07/31 by Prince K. Osei, Prince K Osei, Bernd J. Schroers +1
Mathematics · Physics and Astronomy · #Adjoint representation of a Lie algebra #Advanced Topics in Algebra #Complexification #Homotopy and Cohomology in Algebraic Topology #Killing form #Lie algebra #Lie conformal algebra #Lie group #Noncommutative and Quantum Gravity Theories #Representation of a Lie group #Simple Lie group #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.1063/1.4824704
21 pages, 1 figure, typos corrected
openalex publication_date 2013/10/01 · arxiv created 2013/10/11 · arxiv updated 2015/06/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 Poincaré group in three dimensions. In addition, the construction links the r-matrices with the Bianchi classification of three-dimensional real Lie algebras.