Notes on generalised spin structures

We review some definitions and basic notions relating to generalised spin structures and introduce the notion of reducibility.

We discuss connections on these structures, define a covariant Lie derivative for associated bundles and develop a covariant Cartan calculus.

We introduce an extension of the Lie algebra of Killing vectors, the symmetry algebra, and show that it has a representation on sections of associated bundles.

We discuss homogeneous generalised spin structures and provide a characterisation of them in terms of lifts of the isotropy representation.