Quaternionic vector space

As for vector spaces over any division ring, a quaternionic basis is a family in which every vector has a unique finite right-linear expansion. A finite-dimensional space of quaternionic dimension $n$ is isomorphic to $\mathbb H^n$ and has underlying real dimension $4n$. Right-linear endomorphisms of column vectors in $\mathbb H^n$ are represented by quaternionic matrices acting on the left Givental, Supplement E, pp. 191–193.

Quaternionic conjugation converts a right space $V$ into a left space by defining $q\cdot v=v\overline q$. This is a convention-changing construction, not permission to move scalars through vectors: generally $vq$ and $qv$ are not two notations for the same product. Many geometric texts choose right modules so that matrix groups act from the left; the convention and its use in quaternionic geometry are described in Salamon, §1.5.

The basic example is $\mathbb H^n$ with componentwise right scalar multiplication. Restricting scalars along $\mathbb C\subset\mathbb H$ gives a complex vector space, and then along $\mathbb R\subset\mathbb C$ a real vector space. A real vector space equipped with one complex structure is not thereby quaternionic: it needs compatible actions of quaternionic units $i,j,k$ with $i^2=j^2=k^2=ijk=-1$.

1. Simon Salamon, Riemannian Geometry and Holonomy Groups, Pitman Research Notes in Mathematics 201, Longman Scientific & Technical, 1989. Library catalog record. Relevant: §1.5 on quaternionic linear algebra and conventions used in quaternionic geometry.
2. Alexander Givental, Linear Algebra, Supplement E, “Quaternionic Linear Algebra.” Author course PDF. Relevant: printed pp. 191–193 on handedness, conjugation, bases, and quaternionic matrices.