I am reading a paper that starts talking about 'left vector bundles' and I'm having trouble figuring out what they mean. The specific setup is as follows:
A quarternionic line bundle $L$ over manifold $M$ is a real smooth rank 4 vector bundle with fibers 1-dimensional quaternionic right vector spaces (*).
A complex quarternionic vector bundle is a pair $(L,J)$ with a quaternionic linear endomorphism $J$ such that $J^2=-1$.
A complex quarternionic vector bundle is thus a rank 2 left complex vector bundle whose complex structure is compatible with the right quaternionic structure (**).
(*) Is a right vector space a vector space $V$ with scalar multiplication only defined on the right? So $V \times F \to V$ for field $F$ but $F \times V \to V$ is not defined?
(**) If what I say above is correct, what does the compatibility mean? If this complex quaternionic vector bundle is left, then the compatability means
$a(QJ)=(aQ)J,~a\in F, Q\in \mathbb{H}$
or something? Or does the "rank 2" say the compatability is something like
$\mathbb{H}\oplus J\mathbb{H}=\mathbb{H}J\oplus \mathbb{H}$
The paper I am reading is "Quaternionic Analysis on Riemann Surfaces and Differential Geometry" by Pedit and Pinkall (1998). It seems like my confusion is not related at all to the quaternion structure, would apply to any vector space. Also, I have found some other references to the left- and right- vector bundles in double vector bundles, but that also seems not related to this. Does anyone have any clarity?