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nLab > Latest Changes: G2

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 14th 2012

    added to G2 the definition of G 2G_2 as the subgroup of GL(7)GL(7) that preserves the associative 3-form.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2016
    • (edited Jan 20th 2016)

    Added (here) the characterization of the subgroups of G 2=Aut(𝕆)G_2 = Aut(\mathbb{O}) that stabilize and that fix, respectively, the quaternions ℍ↪𝕆\mathbb{H} \hookrightarrow\mathbb{O}:

    1 = 1 ↓ ↓ Fix G 2(ℍ) ≃ SU(2) ↓ ↓ Stab G 2(ℍ) = Stab G 2(ℍ) ↓ ↓ Aut(ℍ) ≃ SO(3) ↓ ↓ 1 = 1 \array{ 1 &=& 1 \\ \downarrow && \downarrow \\ Fix_{G_2}(\mathbb{H}) & \simeq & SU(2) \\ \downarrow && \downarrow \\ Stab_{G_2}(\mathbb{H}) &= & Stab_{G_2}(\mathbb{H}) \\ \downarrow && \downarrow \\ Aut(\mathbb{H}) &\simeq& SO(3) \\ \downarrow && \downarrow \\ 1 &=& 1 }
    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 20th 2016

    I was wondering if your middle group had another name. Is this saying it is SO(4)SO(4)?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2016
    • (edited Jan 20th 2016)

    Yes, true. Thanks. The source which I had cited also said this, but I forgot to include it. Done now.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 20th 2016

    Added the argument (here) that dim(G 2)=14dim(G_2) = 14 and the argument (here) that Fix G 2(ℍ)≃SU(2)Fix_{G_2}(\mathbb{H}) \simeq SU(2), both using the statement that “octonionic basic triples” form a torsor over G 2G_2, taken from Baez, 4.1.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 30th 2017
    • (edited Oct 30th 2017)

    I added the reference to Basak17, which builds the root space decomposition of the Lie algebra of G 2G_2 from a nice description of the octonions

    Tathagata Basak, Root space decomposition of 𝔤 2\mathfrak{g}_2 from octonions, arXiv:1708.02367

  7. Even a simple dimension count reveals that it cannot possible be the six-sphere.

    Anonymous

    diff, v31, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 12th 2019

    14-8=6

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2019

    I have reverted the edit in revision 31 by “Anonymous” above and put in a link to G2/SU(3) is the 6-sphere

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 14th 2019

    Under Orientation, did you mean to write SO(7)SO(7) instead of SL(7)SL(7)?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 14th 2019

    Thanks, fixed now.

    (Pointer to a reference is missing here, but I don’t have time for it right now.)

    diff, v33, current

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