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Remarks on the structure and integrability of LA-groups

Camilo Angulo

Abstract

We study the structure of an LA-group identifying its underlying VB-group with a representation up to homotopy. We show that the Lie algebroid structure is determined by a complementary action up to homotopy of the Lie algebra of units. We identify the equations that the representation and the action need to verify in order to assemble into an LA-group, establishing an equivalence between LA-groups and LA-matched pairs. As an application, we catalog some extreme examples of representations and actions and comment on their integrability.

Remarks on the structure and integrability of LA-groups

Abstract

We study the structure of an LA-group identifying its underlying VB-group with a representation up to homotopy. We show that the Lie algebroid structure is determined by a complementary action up to homotopy of the Lie algebra of units. We identify the equations that the representation and the action need to verify in order to assemble into an LA-group, establishing an equivalence between LA-groups and LA-matched pairs. As an application, we catalog some extreme examples of representations and actions and comment on their integrability.
Paper Structure (20 sections, 32 theorems, 85 equations)

This paper contains 20 sections, 32 theorems, 85 equations.

Table of Contents

  1. LA -groups and their associated representations up to homotopy
  2. The groupoid structure
  3. Consequences of being LA
  4. Compatibility with the anchors
  5. Compatibility with the brackets
  6. The unit
  7. The source
  8. The target
  9. The multiplication
  10. Applications to structure
  11. LA-Matched Pairs
  12. The Lie algebroid structure
  13. Applications to integrability
  14. The regular case
  15. Vacant groupoids, crossed actions and representations
  16. ...and 5 more sections

Key Result

Theorem 1.1

VBReps Given a VB-group $A\rightrightarrows\mathfrak{h}$ over $G$ with core $\mathfrak{c}$ and a unit extending splitting of its core sequence Eq:CoreSeq, $(\partial,\Delta^\mathfrak{h},\Delta^\mathfrak{c},\Omega)$ is a RUTH of $G$ on $\mathfrak{c}[1]\oplus\mathfrak{h}$. Moreover, the structural map

Theorems & Definitions (67)

  • Theorem 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • proof
  • Remark 1.5
  • Proposition 1.6
  • proof
  • ...and 57 more