Quaternionic vector bundle - Imported Knowlpedia Content

Definition

Let $M$ be a smooth manifold. A quaternionic vector bundle of rank $n$ over $M$ is a smooth vector bundle $\pi:E\to M$ whose fibers are right modules over the quaternion division algebra $\mathbb H$ , together with local trivializations

\Phi_\alpha:E|_{U_\alpha}\longrightarrow U_\alpha\times\mathbb H^n

that are right $\mathbb H$ -linear on every fiber. Equivalently, its transition functions are smooth maps into $\operatorname{GL}(n,\mathbb H)$ . Its underlying real vector bundle has rank $4n$ . Bundle morphisms in this category are smooth, fiberwise right $\mathbb H$ -linear maps.

Equivalent complex description

Restricting scalars along $\mathbb C\subset\mathbb H$ turns $E$ into a complex vector bundle of rank $2n$ . Right multiplication by $j$ defines an antilinear bundle map $J:E\to E$ satisfying $J^2=-I$ . Conversely, such a pair $(E,J)$ recovers the right quaternionic action. This is the standard complex description of a quaternion bundle Atiyah, §1.5.

Metrics and examples

A fiberwise quaternionic Hermitian metric reduces the transition functions from $\operatorname{GL}(n,\mathbb H)$ to the compact symplectic group $\operatorname{Sp}(n)$ . Such a metric can be assembled from local standard metrics by a smooth partition of unity. The product $M\times\mathbb H^n$ is the trivial example, while the tautological line bundle over quaternionic projective space is the basic nontrivial example.

Conventions and near-misses

References

M. F. Atiyah, K-Theory, lecture notes by D. W. Anderson, W. A. Benjamin, 1967. Author-hosted scan. Relevant: §1.5 on quaternion bundles as complex bundles with an antilinear map squaring to $-I$ .
D. Husemoller, Fibre Bundles, 3rd ed., Graduate Texts in Mathematics 20, Springer, 1994. Publisher record. Relevant: Chapters 2–3 on vector bundles, transition functions, and reduction of structure group.