nLab quaternionic structure

Definition

On a complex vector space

A quatermionic structure on a complex vector space $V$ is an complex-anti-linear map

$$\sigma \,\colon\, V \longrightarrow V \,, \;\;\;\;\; \underset{v \in V}{\forall} \; \sigma(\mathrm{i}v) = - \mathrm{i} \sigma(v) \,,$$

which squares to minus the identity:

$$\sigma^2 = -1 \,.$$

(Compare this to a real structure, which is such a $\mathbb{C}$-anti-linear map that instead squares to $+1$, hence that is an involution.)

Given such $\sigma$ then with the linear operator $\mathrm{i}$ of multiplication by the imaginary unit one obtains three real linear endomorphisms

$$\begin{aligned} \mathbf{i} & \coloneqq \mathrm{i} \\ \mathbf{j} & \coloneqq \sigma \\ \mathbf{k} & \coloneqq \mathrm{i} \sigma \end{aligned}$$

on $V$, which evidently induce on $V$ a real linear representation of the algebra of quaternions.

On a manifold

An almost quaternionic structure on a manifold of dimension a multiple of 4, is a reduction of the structure group along the inclusion

$$GL_n(\mathbb{H}) \hookrightarrow Gl_{4 n}(\mathbb{R})$$

of general linear groups, for $\mathbb{R}$ the real numbers and $\mathbb{H}$ the quaternions, hence a G-structure for a quaterionic-general linear group.

(…)

Related concepts

• quaternionic unitary group

• real structure, complex structure

• real vector space, complex vector space, quaternionic vector space

• para-quaternionic structure

References

• Wikipedia, Quaternionic structure

• Wikipedia, Reduction of the structure group – Examples