nLab SU(3)




The special unitary group in 3 complex dimensions.


Subgroups and Supgroups


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:

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.
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
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


  • 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 September 7, 2023 at 12:32:24. See the history of this page for a list of all contributions to it.