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Discovering real lie subalgebras of e6 using Cartan decompositions

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  • The process of complexification is used to classify Lie algebras and identify their Cartan subalgebras. However, this method does not distinguish between real forms of a complex Lie algebra, which can differ in signature. In this paper, we show how Cartan decompositions of a complexified Lie algebra can be combined with information from the Killing form to identify real forms of a given Lie algebra. We apply this technique to sl(3, O), a real form of e(6) with signature (52, 26), thereby identifying chains of real subalgebras and their corresponding Cartan subalgebras within e(6). Motivated by an explicit construction of sl(3, O), we then construct an Abelian group of order 8 which acts on the real forms of e(6), leading to the identification of 8 particular copies of the 5 real forms of e(6), which can be distinguished by their relationship to the original copy of sl(3, O). C (C) 2013 AIP Publishing LLC.
  • This is the publisher’s final pdf. The published article is copyrighted by AIP Publishing LLC and can be found at: http://www.aip.org/
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  • Wangberg, A., & Dray, T. (2013). Discovering real lie subalgebras of e(6) using cartan decompositions. Journal of Mathematical Physics, 54(8) doi:10.1063/1.4818503
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  • 54
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  • 8
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  • The revision was made possible in part through the support of a grant from the John Templeton Foundation.
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