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Spin(5) in nLab

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Contents

Idea

The spin group in dimension 5.

Properties

Exceptional isomorphism

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=ℂ 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 J∈Λ 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 J∈Λ 2V=Λ + 2V⊕Λ − 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

Proposition

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

Consider

  1. the Spin(5)-action on the 4-sphere S 4S^4 which is induced by the defining action on ℝ 5\mathbb{R}^5 under the identification S 4≃S(ℝ 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,ℍ)Spin(5) \simeq Sp(2) = U(2,\mathbb{H}) (from Prop. ) by the canonical left action of U(2,ℍ)U(2,\mathbb{H}) on ℍ 2\mathbb{H}^2 via S 7≃S(ℍ 2)S^7 \simeq S(\mathbb{H}^2).

Then the quaternionic Hopf fibration S 7⟶h ℍS 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

Cohomology

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

Proposition

Let

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 4≃Spin(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 χ∈H 4(BSpin(4),ℤ)\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) k∈H 4(BSpin(5),ℤ), \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) 2∈H 4(BSpin(5),ℤ); \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)=0∈H 4(BSpin(5),ℤ). \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

GG-Structure and exceptional geometry

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References

Last revised on June 23, 2023 at 06:02:33. See the history of this page for a list of all contributions to it.