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arXiv · 1307.6485

Classical r-matrices via semidualisation

Prince K Osei·Bernd J Schroers
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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.

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Prince K Osei, Bernd J Schroers. 2013-07-24. Classical r-matrices via semidualisation. https://doi.org/10.1063/1.4824704

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