On certain classes of $$\mathbf{Sp}(4,\mathbb{R})$$ Sp ( 4 ,
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On certain classes ofΒ ππ©(4, β) symmetric G2 structures PaweΕΒ Nurowski1β Received: 23 April 2020 / Accepted: 28 October 2020 Β© The Author(s) 2020
Abstract We find two different families of ππ©(4, β) symmetric G2 structures in seven dimensions. These are G2 structures with G2 being the split real form of the simple exceptional complex Lie group G2β. The first family has π2 β‘ 0β, while the second family has π1 β‘ π2 β‘ 0β, where π1β, π2 are the celebrated G2-invariant parts of the intrinsic torsion of the G2 structure. The families are different in the sense that the first one lives on a homogeneous space ππ©(4, β)βππ(2, β)lβ, and the second one lives on a homogeneous space ππ©(4, β)βππ(2, β)sβ. Here ππ(2, β)l is an ππ(2, β) corresponding to the π°π©(2, β) related to the long roots in the root diagram of π°π(4, β)β, and ππ(2, β)s is an ππ(2, β) corresponding to the π°π©(2, β) related to the short roots in the root diagram of π°π(4, β). Keywordsβ Homogeneous G2 structuresΒ Β· Skew symmetric torsionΒ Β· Split signature metric
1βIntroduction: aΒ question ofΒ Maciej Dunajski Recently, together with Hill [5], we uncovered an ππ©(4, β) symmetry of the nonholonomic kinematics of a car. I talked about this at the Abel Symposium in Γ
lesund, Norway, in June 2019. After my talk Maciej Dunajski, intrigued by the root diagram of π°π(4, β) which appeared in the talk, asked me if using it I can see a G2 structure on a 7-dimensional homogeneous space M = ππ©(4, β)βππ(2, β)β.
* PaweΕ Nurowski [email protected] 1
Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. LotnikΓ³w 32/46, 02β668Β Warsaw, Poland
13
Vol.:(0123456789)
Annals of Global Analysis and Geometry
My immediate answer was: βI can think about it, but I have to know which of the ππ(2, β) subgroups of ππ©(4, β) I shall use to built M.β The reason for the βbutβ word in my answer was that there are at least two ππ(2, β) subgroups of ππ©(4, β)β, which lie quite differently in there. One can see them in the root diagram above: the first ππ(2, β) corresponds to the long roots, as, for example, E1 and E10β, whereas the second one corresponds to the short roots, as, for example, E2 and E9β. Since Maciej never told me which ππ(2, β) he wants, I decided to consider both of them and to determine what kind of G2 structures one can associate with the respective choice of a subgroup. I emphasize that in the below considerations I will use the split real form of the simple exceptional Lie group G2β. Therefore, the corresponding G2 structure metrics will not be Riemannian.1 They will have signature (3,Β 4).
2βThe Lie algebra π°π(4, β) The Lie algebra π°π(4, β) is given by the 4 Γ 4 matrices
β a5 β βa E = (E π½ ) = β 4 a β 2 ββ2a1 πΌ
a7 a6 a3 a2
a9 a8 βa6 a4
2a10 β a9 β , βa7 β β βa5 β
where the coefficients aI β, I = 1, 2, β¦ 10β, are real constants. The Lie bracket in π°π(4, β) is the usual commutator [E, EοΏ½ ] = E β
EοΏ½ β EοΏ½ β
E of two matrices E and Eβ²β. We start with the following basis (EI ),
EI =
πE , πaI
I = 1, 2, β¦ 10,
in π°π(4, β). In this basis, modulo the antisymmetry, we have the
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