Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: G₂

Bottom of Page

1 to 15 of 15

  1.  
    • 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

    • CommentRowNumber12.
    • CommentAuthorperezl.alonso
    • CommentTimeFeb 7th 2024

    pointer to

    • Guillermo Moreno. The zero divisors of the Cayley-Dickson algebras over the real numbers. (1997) (arXiv:q-alg/9710013)

    where it is shown that the group of zero-divisors of the sedenions is isomorphic to G 2G_2.

    diff, v38, current

    • CommentRowNumber13.
    • CommentAuthorperezl.alonso
    • CommentTimeFeb 7th 2024

    By the way, has the observation in Relation to higher prequantum geometry been used anywhere?

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeFeb 8th 2024

    No, I am not aware that this point of view has been used anywhere.

    • CommentRowNumber15.
    • CommentAuthorSamuel Adrian Antz
    • CommentTimeJul 17th 2024
    • (edited Jul 17th 2024)

    Used unicode subscripts for indices of exceptional Lie groups including title and links. When not linked, usual formulas are used. See discussion here. Links will be re-checked after all titles have been changed. (Removed one redirect for “G2” from the top and added one for “G2” at the bottom of the page.)

    diff, v43, current

1 to 15 of 15

  1.