## 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. (C) 2013 AIP Publishing LLC.

Original language | English |
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Article number | 101702 |

Number of pages | 17 |

Journal | Journal of Mathematical Physics |

Volume | 54 |

Issue number | 10 |

DOIs | |

Publication status | Published - Oct 2013 |

## Keywords

- QUANTUM POINCARE ALGEBRA
- 3 DIMENSIONS
- RELATIVITY
- GRAVITY