Group Theory

Spin geometry



The spin group in dimension 5.


Exceptional isomorphism


There is an exceptional isomorphism

Spin(5)Sp(2) Spin(5) \;\simeq\; Sp(2)

between Spin(5) and the quaternionic unitary group Sp(2)=U(2,)Sp(2) = U(2,\mathbb{H}).

This is an indirect consequence of triality, see e.g. Čadek-Vanžura 97. Alternatively, it can be shown as follows.

Let VV be a 4-dimensional complex vector space with an inner product and a compatible complex volume form. As explained here, this structure can be used to define a conjugate-linear Hodge star operator on Λ 2V\Lambda^2 V whose +1+1 and 1-1 eigenspaces, say Λ ± 2V\Lambda_{\pm}^2 V, are each 6-dimensional real inner product spaces. Thus, the group SU(V)\mathrm{SU}(V) acts as linear transformations of Λ ± 2V\Lambda_{\pm}^2 V that preserve the inner product, giving a homomorphism ρ:SU(V)O(Λ ± 2V)\rho: \mathrm{SU}(V) \to \mathrm{O}(\Lambda_{\pm}^2 V). In fact ρ\rho maps SU(V)\mathrm{SU}(V) in a 2-1 and onto way to SO(Λ ± 2V)\mathrm{SO}(\Lambda_{\pm}^2 V). Taking V= 4V = \mathbb{C}^4 this shows SU(4)Spin(6)\mathrm{SU}(4) \cong \mathrm{Spin}(6).

Now suppose VV is additionally equipped with an complex symplectic structure, i.e. a nondegenerate skew-symmetric complex-bilinear form JΛ 2VJ \in \Lambda^2 V. The subgroup Sp(V)\mathrm{Sp}(V) of SU(V)\mathrm{SU}(V) preserving this extra structure is isomorphic to the compact symplectic group Sp(2)\mathrm{Sp}(2), which is also the quaternionic unitary group. This subgroup Sp(V)\mathrm{Sp}(V) acts on Λ + 2V\Lambda_+^2 V and Λ 2V\Lambda_-^2 V preserving JΛ 2V=Λ + 2VΛ 2VJ \in \Lambda^2 V = \Lambda_+^2 V \oplus \Lambda_-^2 V. Since Sp(V)\mathrm{Sp}(V) is compact, every invariant subspace has an invariant complement, so one or both of the 6-dimensional subspaces Λ + 2V\Lambda_+^2 V and Λ 2V\Lambda_-^2 V must have a 5-dimensional subspace invariant under the action of Sp(V)\mathrm{Sp}(V). This shows that the double cover ρ:SU(4)SO(6)\rho: \mathrm{SU}(4) \to \mathrm{SO}(6) restricts to a 2-1 homomorphism σ:Sp(2)SO(5)\sigma : \mathrm{Sp}(2) \to \mathrm{SO}(5). Since

dim(Sp(2))=10=dim(SO(5)) \dim(\mathrm{Sp}(2)) = 10 = \dim(\mathrm{SO}(5))

the differential dσd\sigma, being injective, must also be surjective. Thus σ:Sp(2)SO(5)\sigma : \mathrm{Sp}(2) \to \mathrm{SO}(5) is actually a double cover. Since Sp(2)\mathrm{Sp}(2) is connected this implies Sp(2)Spin(5)\mathrm{Sp}(2) \cong \mathrm{Spin}(5).

Action on quaternionic Hopf fibration


(Spin(5)-equivariance of quaternionic Hopf fibration)


  1. the Spin(5)-action on the 4-sphere S 4S^4 which is induced by the defining action on 5\mathbb{R}^5 under the identification S 4S( 5)S^4 \simeq S(\mathbb{R}^5);

  2. the Spin(5)-action on the 7-sphere S 7S^7 which is induced under the exceptional isomorphism Spin(5)Sp(2)=U(2,)Spin(5) \simeq Sp(2) = U(2,\mathbb{H}) (from Prop. ) by the canonical left action of U(2,)U(2,\mathbb{H}) on 2\mathbb{H}^2 via S 7S( 2)S^7 \simeq S(\mathbb{H}^2).

Then the quaternionic Hopf fibration S 7h S 4S^7 \overset{h_{\mathbb{H}}}{\longrightarrow} S^4 is equivariant with respect to these actions.

This is almost explicit in Porteous 95, p. 263



The integral cohomology ring of the classifying space BSpin(5)B Spin(5) is spanned by two generators

  1. the first fractional Pontryagin class 12p 1\tfrac{1}{2}p_1

  2. the linear combination 12p 212(p 1) 2\tfrac{1}{2}p_2 - \tfrac{1}{2}(p_1)^2 of the half the second Pontryagin class with half the cup product-square of the first Pontryagin class:

H (BSpin(5),)[12p 1,12p 212(p 1) 2] H^\bullet \big( B Spin(5), \mathbb{Z} \big) \;\simeq\; \mathbb{Z} \left[ \tfrac{1}{2}p_1, \; \tfrac{1}{2}p_2 - \tfrac{1}{2}(p_1)^2 \right]

This is a special case of the general statement in Pittie 91, see e.g. Kalkkinen 06, Section 3).



S 4 BSpin(4) π BSpin(5) \array{ S^4 &\longrightarrow& B Spin(4) \\ && \big\downarrow^{\mathrlap{\pi}} \\ && B Spin(5) }

be the spherical fibration of classifying spaces induced from the canonical inclusion of Spin(4) into Spin(5) and using that the 4-sphere is equivalently the coset space S 4Spin(5)/Spin(4)S^4 \simeq Spin(5)/Spin(4) (this Prop.).

Then the fiber integration of the odd cup powers χ 2k+1\chi^{2k+1} of the Euler class χH 4(BSpin(4),)\chi \in H^4\big( B Spin(4), \mathbb{Z}\big) (see this Prop) are proportional to cup powers of the second Pontryagin class

π *(χ 2k+1)=2(p 2) kH 4(BSpin(5),), \pi_\ast \left( \chi^{2k+1} \right) \;=\; 2 \big( p_2 \big)^k \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,,

for instance

π *(χ) =2 π *(χ 3) =2p 2 π *(χ 5) =2(p 2) 2H 4(BSpin(5),); \begin{aligned} \pi_\ast \big( \chi \big) & = 2 \\ \pi_\ast \left( \chi^3 \right) & = 2 p_2 \\ \pi_\ast \left( \chi^5 \right) & = 2 (p_2)^2 \end{aligned} \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,;

while the fiber integration of the even cup powers χ 2k\chi^{2k} vanishes

π *(χ 2k)=0H 4(BSpin(5),). \pi_\ast \left( \chi^{2k} \right) \;=\; 0 \;\;\in\;\; H^4\big( B Spin(5), \mathbb{Z} \big) \,.

(Bott-Cattaneo 98, Lemma 2.1)

Coset spaces

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

rotation groups in low dimensions:

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group

see also



Last revised on January 16, 2021 at 00:41:19. See the history of this page for a list of all contributions to it.