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On n-tuple principal bundles

Katarzyna Grabowska, Janusz Grabowski

Abstract

We develop the concept of a double (more generally n-tuple) principal bundle departing from a compatibility condition for a principal action of a Lie group on a groupoid.

On n-tuple principal bundles

Abstract

We develop the concept of a double (more generally n-tuple) principal bundle departing from a compatibility condition for a principal action of a Lie group on a groupoid.

Paper Structure

This paper contains 11 sections, 17 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. $G$-groupoids
  3. The compatibility condition
  4. Structure of $G$-groupoids
  5. Double principal bundles
  6. Double principal groups and their actions
  7. $n$-tuple principal groups and bundles
  8. Associated bundles and principal connections
  9. Associated double bundles
  10. Frame bundle of a double vector bundle
  11. Higher double graded bundles

Key Result

Proposition 2.3

(cf. Bruce:2015) A compatible action $焚:G\times\mathcal{G}\to\mathcal{G}$ of a Lie group $G$ on a Lie groupoid $\mathcal{G}\rightrightarrows M$ induces a canonical action $焚_M:G\times M\to M$ on the manifold $M$ of units. If $焚$ is principal, then $焚_M$ is also principal. In particular, in this case

Theorems & Definitions (41)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.1
  • Remark 3.2
  • ...and 31 more