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Reductive Homogeneous Spaces of the Compact Lie Group \(G_2\)

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Non-Associative Algebras and Related Topics (NAART 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 427))

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Abstract

The first author defended her doctoral thesis “Espacios homogéneos reductivos y álgebras no asociativas” in 2001, supervised by P. Benito and A. Elduque. This thesis contained the classification of the Lie-Yamaguti algebras with standard envelo** algebra \(\mathfrak {g}_2\) over fields of characteristic zero, which in particular gives the classification of the homogeneous reductive spaces of the compact Lie group \(G_2\). In this work we revisit this classification from a more geometrical approach. We provide too geometric models of the corresponding homogeneous spaces and make explicit some relations among them.

Partially supported by Junta de Andalucía projects FQM-336, UMA18-FEDERJA-119 and P20_01391, and by the Spanish MICINN projects PID2019-104236GB-I00/AEI/10.13039/501100011033 and PID2020-118452GB-I00, all of them with FEDER funds.

Partially supported by Spanish MICINN project PID2020-118452GB-I00 and Andalusian and ERDF projects FQM-494 and P20\(_{-}\)01391.

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Notes

  1. 1.

    Note that l and r correspond to − and \(+\), respectively, in the literature.

  2. 2.

    Be careful with the typo in that formula in [39, Proposition 2.4].

  3. 3.

    Note that, if \(J^2=-\textrm{id}\), then \(J\in \textrm{SO}(W)\) if and only if \(J\in \mathfrak {so}(W)\).

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Authors and Affiliations

  1. Universidad de Málaga, Málaga, Spain

    Cristina Draper & Francisco J. Palomo

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  1. Cristina Draper
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  2. Francisco J. Palomo
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Correspondence to Cristina Draper .

Editor information

Editors and Affiliations

  1. Mathematics, University of Coimbra, Coimbra, Portugal

    Helena Albuquerque

  2. Department of Mathematics, Centre for Mathematics of the University of Coimbra, Coimbra, Portugal

    Jose Brox

  3. Mathematics, University of Oviedo, Oviedo, Spain

    Consuelo Martínez

  4. CMUC-Centre for Mathematics of the University of Coimbra, CeBER-Centre for Business and Economics Research, Faculty of Economics, University of Coimbra, Coimbra, Portugal

    Paulo Saraiva

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Draper, C., Palomo, F.J. (2023). Reductive Homogeneous Spaces of the Compact Lie Group \(G_2\). In: Albuquerque, H., Brox, J., Martínez, C., Saraiva, P. (eds) Non-Associative Algebras and Related Topics. NAART 2020. Springer Proceedings in Mathematics & Statistics, vol 427. Springer, Cham. https://doi.org/10.1007/978-3-031-32707-0_3

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