nLab
G2

Contents

Context

Exceptional structures

Group Theory

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

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.

Definition

Definition

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 \,,

where

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).

Properties

Orientation

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

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

Dimension

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.

Hence

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)

Cohomology

The Dwyer-Wilkerson space G 3G_3 (Dwyer-Wilkerson 93) is a 2-complete H-space, in fact a finite loop space/infinity-group, such that the mod 2 cohomology ring of its classifying space/delooping is the mod 2 Dickson invariants of rank 4. As such, it is the fourth and last space in a series of infinity-groups that starts with 3 compact Lie groups:

n=n=1234
DI(n)=DI(n)=Z/2SO(3)G2G3
= Aut(C)= Aut(H)= Aut(O)

Subgroups

We discuss various subgroups of G 2G_2.

Definition

Write

  • 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}}.

Proposition

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

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

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).

Proposition

The subgroups in def. 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)

The coset space G2/SU(3) is the 6-sphere. See there for more.

(from Kramer 02)

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

\,

Supgroups

Proposition

(coset space of Spin(7) by G2 is 7-sphere)

Consider the canonical action of Spin(7) on the unit sphere in 8\mathbb{R}^8 (the 7-sphere),

  1. This action is is transitive;

  2. the stabilizer group of any point on S 7S^7 is G2;

  3. all G2-subgroups of Spin(7) arise this way, and are all conjugate to each other.

Hence the coset space of Spin(7) by G2 is the 7-sphere

Spin(7)/G 2S 7. Spin(7)/G_2 \;\simeq\; S^7 \,.

(e.g Varadarajan 01, Theorem 3)

Coset quotients

coset space-structures on n-spheres:

standard:
S n1 diffSO(n)/SO(n1)S^{n-1} \simeq_{diff} SO(n)/SO(n-1)this Prop.
S 2n1 diffSU(n)/SU(n1)S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)this Prop.
S 4n1 diffSp(n)/Sp(n1)S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)this Prop.
exceptional:
S 7 diffSpin(7)/G 2S^7 \simeq_{diff} Spin(7)/G_2Spin(7)/G2 is the 7-sphere
S 7 diffSpin(6)/SU(3)S^7 \simeq_{diff} Spin(6)/SU(3)since Spin(6) \simeq SU(4)
S 7 diffSpin(5)/SU(2)S^7 \simeq_{diff} Spin(5)/SU(2)since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere
S 6 diffG 2/SU(3)S^6 \simeq_{diff} G_2/SU(3)G2/SU(3) is the 6-sphere
S 15 diffSpin(9)/Spin(7)S^15 \simeq_{diff} Spin(9)/Spin(7)Spin(9)/Spin(7) is the 15-sphere

see also Spin(8)-subgroups and reductions

homotopy fibers of homotopy pullbacks of classifying spaces:

(from FSS 19, 3.4)

GG-Structure and exceptional geometry

Spin(8)-subgroups and reductions to exceptional geometry

reductionfrom spin groupto maximal subgroup
Spin(7)-structureSpin(8)Spin(7)
G2-structureSpin(7)G2
CY3-structureSpin(6)SU(3)
SU(2)-structureSpin(5)SU(2)
generalized reductionfrom Narain groupto direct product group
generalized Spin(7)-structureSpin(8,8)Spin(8,8)Spin(7)×Spin(7)Spin(7) \times Spin(7)
generalized G2-structureSpin(7,7)Spin(7,7)G 2×G 2G_2 \times G_2
generalized CY3Spin(6,6)Spin(6,6)SU(3)×SU(3)SU(3) \times SU(3)

see also: coset space structure on n-spheres

Relation to higher prequantum geometry

The 3-form ω\omega from def. 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. 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-structure\,\,special holonomy\,\,dimension\,\,preserved differential form\,
\,\mathbb{C}\,\,Kähler manifold\,\,U(n)\,2n\,2n\,\,Kähler forms ω 2\omega_2\,
\,Calabi-Yau manifold\,\,SU(n)\,2n\,2n\,
\,\mathbb{H}\,\,quaternionic Kähler manifold\,\,Sp(n).Sp(1)\,4n\,4n\,ω 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 manifold\,\,Sp(n)\,4n\,4n\,ω=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) manifold\,\,Spin(7)\,\,8\,\,Cayley form\,
\,G2 manifold\,\,G2\,7\,7\,\,associative 3-form\,

References

General

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

Discussion of G 2G_2 as a subgroup of Spin(7):

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)

Last revised on August 25, 2019 at 10:37:26. See the history of this page for a list of all contributions to it.