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The Lie group G 2G_2 is one (or rather: three) of the exceptional Lie groups. One way to characterize it is as the automorphism group of the octonions as a normed algebra:

G 2=Aut(𝕆). G_2 = Aut(\mathbb{O}) \,.

Another way to characterize it is as the stabilizer subgroup inside the general linear group GL(7)GL(7) of the canonical differential 3-form ,()×()\langle ,(-)\times (-) \rangle on the Cartesian space 7\mathbb{R}^7

G 2Stab GL(7)(,×). G_2 \simeq Stab_{GL(7)}(\langle -, -\times -\rangle) \,.

As such, the group G 2G_2 is a higher analog of the symplectic group (which is the group that preserves a canonical 2-form on any 2n\mathbb{R}^{2n}), obtained by passing from symplectic geometry to 2-plectic geometry.



On the Cartesian space 7\mathbb{R}^7 consider the associative 3-form, the constant differential 3-form ωΩ 3( 7)\omega \in \Omega^3(\mathbb{R}^7) given on tangent vectors u,v,w 7u,v,w \in \mathbb{R}^7 by

ω(u,v,w)u,v×w, \omega(u,v,w) \coloneqq \langle u , v \times w\rangle \,,


Then the group G 2GL(7)G_2 \hookrightarrow GL(7) is the subgroup of the general linear group acting on 7\mathbb{R}^7 which preserves the canonical orientation and preserves this 3-form ω\omega. Equivalently, it is the subgroup preserving the orientation and the Hodge dual differential 4-form ω\star \omega.

See for instance the introduction of (Joyce).



The inclusion G 2GL(7)G_2 \hookrightarrow GL(7) of def. 1 factors through the special orthogonal group

G 2SL(7)GL(7). G_2 \hookrightarrow SL(7) \hookrightarrow GL(7) \,.


The dimension of (the manifold underlying) G 2G_2 is

dim(G 2)=14. dim(G_2) = 14 \,.

One way to see this is via octonionic basic triples (e 1,e 2,e 3)𝕆 3(e_1, e_2, e_3) \in \mathbb{O}^3 and the fact (this proposition) that these form a torsor over G 2G_2, hence that the space of them has the same dimension as G 2G_2:

  • the space of choices for e 1e_1 is the 6-sphere of imaginary unit octonions;

  • given that, the space of choices for e 2e_2 is a 5-sphere of imaginary unit octonions orthogonal to e 1e_1;

  • given that, then the space of choices for e 3e_3 is the 3-sphere of imaginary unit octonions orthogonal to both e 1e_1 and e 2e_2.


dim(G 2)=dim(S 6)+dim(S 5)+dim(S 3)=14. dim(G_2) = dim(S^6) + dim(S^5) + dim(S^3) = 14 \,.

(e.g. Baez, 4.1)


We discuss various subgroups of G 2G_2.



  • G 2=Aut(𝕆)G_2 = Aut(\mathbb{O}), the automorphism group of the octonions as a normed alegbra,

  • Stab G 2()Stab_{G_2}(\mathbb{H}), the stabilizer subgroup of the quaternions inside the octonions, i.e. of elements σG 2\sigma\in G_2 such that σ |:𝕆\sigma_{|\mathbb{H}}\colon \mathbb{H}\to \mathbb{H} \hookrightarrow\mathbb{O};

  • Fix G 2()Fix_{G_2}(\mathbb{H}) for the further subgroup of elements that fix each quaternions (the “elementwise stabilizer group”), i.e. those σ\sigma with σ |=id \sigma_{\vert \mathbb{H}} = id_{\mathbb{H}}.


The elementwise stabilizer group of the quaternions is SU(2):

Fix G 2()SU(2). Fix_{G_2}(\mathbb{H}) \simeq SU(2) \,.

Consider octonionic basic triples (e 1,e 2,e 3)𝕆 3(e_1, e_2, e_3) \in \mathbb{O}^3 and the fact (this proposition) that these form a torsor over G 2G_2.

The choice of (e 1,e 2)(e_1,e_2) is equivalently a choice of inclusion 𝕆\mathbb{H} \hookrightarrow \mathbb{O}. Then the remaining space of choices for e 3e_3 is the 3-sphere (the space of unit imaginary octonions orthogonal to both e 1e_1 and e 2e_2). This carries a unit group structure, and by the torsor property this is the required subgroup of SU(2)SU(2).


The subgroups in def. 2 sit in a short exact sequence of the form

1 = 1 Fix G 2() SU(2) Stab G 2() SO(4) Aut() SO(3) 1 = 1 \array{ 1 &=& 1 \\ \downarrow && \downarrow \\ Fix_{G_2}(\mathbb{H}) & \simeq & SU(2) \\ \downarrow && \downarrow \\ Stab_{G_2}(\mathbb{H}) & \simeq & SO(4) \\ \downarrow && \downarrow \\ Aut(\mathbb{H}) &\simeq& SO(3) \\ \downarrow && \downarrow \\ 1 &=& 1 }

exhibiting SO(4) as a group extension of the special orthogonal group SO(3)SO(3) by the special unitary group SU(2)SU(2).

(e.g. Ferolito, section 4)

Furthermore there is a subgroup SU(3)G 2SU(3) \hookrightarrow G_2 whose intersection with SO(4)SO(4) is U(2)U(2). The simple part SU(2)SU(2) of this intersection is a normal subgroup of SO(4)SO(4).

(see e.g. Miyaoka 93)

(from Kramer 02)

The Weyl group of G 2G_2 is the dihedral group of order 12. (see e.g. Ishiguro, p. 3).

Relation to higher prequantum geometry

The 3-form ω\omega from def. 1 we may regard as equipping 7\mathbb{R}^7 with 2-plectic structure. From this point of view G 2G_2 is the linear subgroup of the 2-plectomorphism group, hence (up to the translations) the image of the Heisenberg group of ( 7,ω)(\mathbb{R}^7, \omega) in the symplectomorphism group.

Or, dually, we may regard the 4-form ω\star \omega of def. 1 as being a 3-plectic structure and G 2G_2 correspondingly as the linear part in the 3-plectomorphism group of 7\mathbb{R}^7.

classification of special holonomy manifolds by Berger's theorem:

G-structurespecial holonomydimensionpreserved differential form
\mathbb{C}Kähler manifoldU(k)2k2kKähler forms ω 2\omega_2
Calabi-Yau manifoldSU(k)2k2k
\mathbb{H}quaternionic Kähler manifoldSp(k)Sp(1)4k4kω 4=ω 1ω 1+ω 2ω 2+ω 3ω 3\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3
hyper-Kähler manifoldSp(k)4k4kω=aω 2 (1)+bω 2 (2)+cω 2 (3)\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2 (a 2+b 2+c 2=1a^2 + b^2 + c^2 = 1)
𝕆\mathbb{O}Spin(7) manifoldSpin(7)8Cayley form
G2 manifoldG277associative 3-form



Surveys are in

  • Spiro Karigiannis, What is… a G 2G_2-manifold (pdf)

  • Simon Salamon, A tour of exceptional geometry, (pdf)

  • Wikipedia, G2 .

The definitions are reviewed for instance in

  • Dominic Joyce, Compact Riemannian 7-manifolds with holonomy G 2G_2, Journal of Differential Geometry vol 43, no 2 (pdf)

  • Ferolito The octonions and G 2G_2 (pdf)

  • John Baez, section 4.1 G2, of The Octonions (arXiv:math/0105155)

  • Ruben Arenas, Constructing a Matrix Representation of the Lie Group G 2G_2, 2005 (pdf)

Discussion in terms of the Heisenberg group in 2-plectic geometry is in

A description of the root space decomposition of the Lie algebra 𝔤 2\mathfrak{g}_2 is in

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

Cohomological properties are discussed in

  • Younggi Choi, Homology of the gauge group of exceptional Lie group G 2G_2, J. Korean Math. Soc. 45 (2008), No. 3, pp. 699–709

Discussion of subgroups includes

  • Reiko Miyaoka, The linear isotropy group of G 2/SO(4)G_2/SO(4), the Hopf fibering and isoparametric hypersurfaces, Osaka J. Math. Volume 30, Number 2 (1993), 179-202. (Euclid)

  • Kenshi Ishiguro, Classifying spaces and a subgroup of the exceptional Lie group G 2G_2 pdf

  • Linus Kramer, 4.27 of Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurfaces, AMS 2002

Applications in physics

Discussion of Yang-Mills theory with G 2G_2 as gauge group is in

  • Ernst-Michael Ilgenfritz, Axel Maas, Topological aspects of G 2G_2 Yang-Mills theory (arXiv:1210.5963)
Revised on October 30, 2017 00:04:14 by David Roberts (