The special unitary group in 3 complex dimensions.
We have the following commuting diagram of subgroup inclusions, where each square exhibits a pullback/fiber product, hence an intersection of subgroups:
Here in the bottom row we have the Lie groups
Spin(5)$\hookrightarrow$ Spin(6) $\hookrightarrow$ Spin(7) $\hookrightarrow$ Spin(8)
and in the top row we have
SU(2)$\hookrightarrow$ SU(3) $\hookrightarrow$ G2 $\hookrightarrow$ Spin(7)
This is a re-statement of Onishchik 93, Table 2, p. 144:
coset space-structures on n-spheres:
standard: | |
---|---|
$S^{n-1} \simeq_{diff} SO(n)/SO(n-1)$ | this Prop. |
$S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)$ | this Prop. |
$S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)$ | this Prop. |
exceptional: | |
$S^7 \simeq_{diff} Spin(7)/G_2$ | Spin(7)/G2 is the 7-sphere |
$S^7 \simeq_{diff} Spin(6)/SU(3)$ | since Spin(6) $\simeq$ SU(4) |
$S^7 \simeq_{diff} Spin(5)/SU(2)$ | since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere |
$S^6 \simeq_{diff} G_2/SU(3)$ | G2/SU(3) is the 6-sphere |
$S^15 \simeq_{diff} Spin(9)/Spin(7)$ | 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)
Spin(8)-subgroups and reductions to exceptional geometry
reduction | from spin group | to maximal subgroup |
---|---|---|
Spin(7)-structure | Spin(8) | Spin(7) |
G2-structure | Spin(7) | G2 |
CY3-structure | Spin(6) | SU(3) |
SU(2)-structure | Spin(5) | SU(2) |
generalized reduction | from Narain group | to direct product group |
generalized Spin(7)-structure | $Spin(8,8)$ | $Spin(7) \times Spin(7)$ |
generalized G2-structure | $Spin(7,7)$ | $G_2 \times G_2$ |
generalized CY3 | $Spin(6,6)$ | $SU(3) \times SU(3)$ |
see also: coset space structure on n-spheres
A. L. Onishchik (ed.) Lie Groups and Lie Algebras
I. A. L. Onishchik, E. B. Vinberg, Foundations of Lie Theory,
II. V. V. Gorbatsevich, A. L. Onishchik, Lie Transformation Groups
Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993
Last revised on August 16, 2020 at 00:40:04. See the history of this page for a list of all contributions to it.