Notes on generalised spin structures

TL;DR

This paper reviews generalized spin structures, introduces reducibility, and develops a covariant calculus, expanding the understanding of symmetry and connections in geometric structures.

Contribution

It introduces the notion of reducibility for generalized spin structures and develops a covariant Cartan calculus and symmetry algebra representations.

Findings

Defined reducibility for generalized spin structures

Developed a covariant Lie derivative and Cartan calculus

Characterized homogeneous generalized spin structures

Abstract

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.