nLab SU(3)

Contents

Contents

Idea

The special unitary group in 3 complex dimensions.

Properties

Subgroups and Supgroups

Proposition

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 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

References

  • 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

  • Howard Georgi, §7 & §9 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]

    with an eye towards application to (the standard model of) particle physics

Last revised on July 18, 2024 at 11:20:15. See the history of this page for a list of all contributions to it.