Cohomology and Extensions
∞-Lie theory (higher geometry)
Formal Lie groupoids
The Lie group 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:
Another way to characterize it is as the stabilizer subgroup inside the general linear group of the canonical differential 3-form on the Cartesian space
As such, the group is a higher analog of the symplectic group (which is the group that preserves a canonical 2-form on any ), obtained by passing from symplectic geometry to 2-plectic geometry.
On the Cartesian space consider the associative 3-form, the constant differential 3-form given on tangent vectors by
Then the group is the subgroup of the general linear group acting on which preserves the canonical orientation and preserves this 3-form . Equivalently, it is the subgroup preserving the orientation and the Hodge dual differential 4-form .
See for instance the introduction of (Joyce).
The inclusion of def. 1 factors through the special orthogonal group
The dimension of (the manifold underlying) is
One way to see this is via octonionic basic triples and the fact (this proposition) that these form a torsor over , hence that the space of them has the same dimension as :
the space of choices for is the 6-sphere of imaginary unit octonions;
given that, the space of choices for is a 5-sphere of imaginary unit octonions orthogonal to ;
given that, then the space of choices for is the 3-sphere of imaginary unit octonions orthogonal to both and .
(e.g. Baez, 4.1)
We discuss various subgroups of .
, the automorphism group of the octonions as a normed alegbra,
, the stabilizer subgroup of the quaternions inside the octonions, i.e. of elements such that ;
for the further subgroup of elements that fix each quaternions (the “elementwise stabilizer group”), i.e. those with .
The elementwise stabilizer group of the quaternions is SU(2):
Consider octonionic basic triples and the fact (this proposition) that these form a torsor over .
The choice of is equivalently a choice of inclusion . Then the remaining space of choices for is the 3-sphere (the space of unit imaginary octonions orthogonal to both and ). This carries a unit group structure, and by the torsor property this is the required subgroup of .
The subgroups in def. 2 sit in a short exact sequence of the form
exhibiting SO(4) as a group extension of the special orthogonal group by the special unitary group .
(e.g. Ferolito, section 4)
Furthermore there is a subgroup whose intersection with is . The simple part of this intersection is a normal subgroup of .
(see e.g. Miyaoka 93)
(from Kramer 02)
The Weyl group of is the dihedral group of order 12. (see e.g. Ishiguro, p. 3).
Relation to higher prequantum geometry
The 3-form from def. 1 we may regard as equipping with 2-plectic structure. From this point of view is the linear subgroup of the 2-plectomorphism group, hence (up to the translations) the image of the Heisenberg group of in the symplectomorphism group.
Or, dually, we may regard the 4-form of def. 1 as being a 3-plectic structure and correspondingly as the linear part in the 3-plectomorphism group of .
G2 manifold, generalized G2-manifold
M-theory on G2-manifolds, G2-MSSM
E6, E7, E8, E9, E10, E11,
classification of special holonomy manifolds by Berger's theorem:
Surveys are in
Spiro Karigiannis, What is… a -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 , Journal of Differential Geometry vol 43, no 2 (pdf)
Ferolito The octonions and (pdf)
John Baez, section 4.1 G2, of The Octonions (arXiv:math/0105155)
Ruben Arenas, Constructing a Matrix Representation of the Lie Group , 2005 (pdf)
Discussion in terms of the Heisenberg group in 2-plectic geometry is in
Cohomological properties are discussed in
- Younggi Choi, Homology of the gauge group of exceptional Lie group , J. Korean Math. Soc. 45 (2008), No. 3, pp. 699–709
Discussion of subgroups includes
Reiko Miyaoka, The linear isotropy group of , 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 pdf
Linus Kramer, 4.27 of Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurfaces, AMS 2002
Applications in physics
Discussion of Yang-Mills theory with as gauge group is in
- Ernst-Michael Ilgenfritz, Axel Maas, Topological aspects of Yang-Mills theory (arXiv:1210.5963)