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DOMOSCHOOL 2018: Einstein Equations: Physical and Mathematical Aspects of General Relativity pp 93–137Cite as

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The Bianchi Classification of the Three-Dimensional Lie Algebras and Homogeneous Cosmologies and the Mixmaster Universe

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Abstract

The Bianchi classification of the three-dimensional Lie algebras and of the spatially homogeneous universes is presented in a rather pedagogical manner. The dynamics of the Bianchi-I, Bianchi-II and Bianchi-IX universes are discussed in detail. A special attention is paid to the phenomenon of the oscillatory approach to the cosmological singularity (BKL) known also as the Mixmaster universe. The stochasticity in cosmology and the connection between cosmological billiards and infinite-dimensional Lie algebras are briefly illustrated.

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Author information

Authors and Affiliations

  1. Department of Physics and Astronomy of the University of Bologna and INFN, Bologna, Italy

    Alexander Yu. Kamenshchik

  2. L.D. Landau Institute for Theoretical Physics, Moscow, Russia

    Alexander Yu. Kamenshchik

Authors
  1. Alexander Yu. Kamenshchik
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Correspondence to Alexander Yu. Kamenshchik .

Editor information

Editors and Affiliations

  1. Department of Science and High Technology, University Insubrien, Como, Italy

    Sergio Cacciatori

  2. Mathematisches Institut, Humboldt-Universität zu Berlin, Berlin, Germany

    Batu Güneysu

  3. Department of Science and High Technology, University Insubrien, Como, Italy

    Stefano Pigola

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Kamenshchik, A.Y. (2019). The Bianchi Classification of the Three-Dimensional Lie Algebras and Homogeneous Cosmologies and the Mixmaster Universe. In: Cacciatori, S., Güneysu, B., Pigola, S. (eds) Einstein Equations: Physical and Mathematical Aspects of General Relativity. DOMOSCHOOL 2018. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-18061-4_3

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