Atiyah constructions for Lie algebroid connections on fiber bundles

TL;DR

This paper develops a generalized Atiyah algebroid framework to unify Lie algebroid connections on vector and principal bundles, providing explicit constructions and examples to characterize key connection properties.

Contribution

It introduces a unified Atiyah algebroid structure for Lie algebroid connections, extending classical concepts to encompass both vector and principal bundles with explicit constructions.

Findings

Defined a generalized Atiyah algebroid structure

Constructed explicit Atiyah-type extensions and sequences

Applied framework to holomorphic and invariant connections

Abstract

To address the need for a unified framework that incorporates Lie algebroid connections on both vector and principal bundles, this paper investigates a generalized Atiyah algebroid structure and its short exact sequence. Building on this generalization, we describe Atiyah-type extensions and sequences that represent Atiyah classes through three explicit constructions designed to encode Lie algebroid connections compatible with specified sub-structures. As illustrative examples, we work out an enriched Atiyah algebroid construct, providing a systematic tool to characterize certain key properties of holomorphic connections and invariant connections.