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Notes on relative algebroids

Wilmer Smilde

TL;DR

The notes introduce relative algebroids as a framework unifying Lie algebroids with PDEs, emphasizing prolongation and formal integrability and their relation to PDEs with symmetries. They develop the structural theory of relative derivations, define relative algebroids, and explain how prolongations produce (profinite) towers that control realizations of geometric classification problems. By connecting to jet-space PDEs and symmetry quotients, the work shows how relative algebroids capture both finite- and infinite-type classification phenomena and provide a route to realize solutions via Cartan–Kaehler type methods. The discussions are reinforced with concrete examples, including the relative algebroid structure underlying surfaces with prescribed curvature behavior and PDE quotients by symmetries, highlighting potential for broader geometric and analytical applications.

Abstract

Relative algebroids provide a framework that unifies Lie algebroids with partial differential equations. In this set of notes, we explain how relative algebroids arise from geometric problems, and give an introduction to their structural theory. We also discuss their relation to and relevance for partial differential equations with symmetry.

Notes on relative algebroids

TL;DR

The notes introduce relative algebroids as a framework unifying Lie algebroids with PDEs, emphasizing prolongation and formal integrability and their relation to PDEs with symmetries. They develop the structural theory of relative derivations, define relative algebroids, and explain how prolongations produce (profinite) towers that control realizations of geometric classification problems. By connecting to jet-space PDEs and symmetry quotients, the work shows how relative algebroids capture both finite- and infinite-type classification phenomena and provide a route to realize solutions via Cartan–Kaehler type methods. The discussions are reinforced with concrete examples, including the relative algebroid structure underlying surfaces with prescribed curvature behavior and PDE quotients by symmetries, highlighting potential for broader geometric and analytical applications.

Abstract

Relative algebroids provide a framework that unifies Lie algebroids with partial differential equations. In this set of notes, we explain how relative algebroids arise from geometric problems, and give an introduction to their structural theory. We also discuss their relation to and relevance for partial differential equations with symmetry.
Paper Structure (27 sections, 9 theorems, 119 equations)

This paper contains 27 sections, 9 theorems, 119 equations.

Table of Contents

  1. Motivating examples and overview
  2. Geometric structures and coframes
  3. Appearances of algebroids in classification problems for geometric structures
  4. From algebroids to classification
  5. Finite type classification problems
  6. Infinite type classification problems
  7. Relative derivations
  8. Derivations, brackets and algebroids
  9. The structure of derivations
  10. Lie algebroids
  11. Relative derivations and brackets
  12. Structure of relative derivations
  13. Relative algebroids
  14. Prolongation
  15. Computing prolongations
  16. ...and 12 more sections

Key Result

Lemma 3.4

Let $(A, p, {\mathrm{D}})$ be a relative algebroid relative to $p\colon M\to N$. Then there is a map $c_{\mathrm{D}}\colon M\to \mathcal{D}^1_A$ such that $p = p_1\circ c_{\mathrm{D}}$ and ${\mathrm{D}} = c^*_{\mathrm{D}} \Breve{\mathrm{D}}$. Conversely, given any map $c\colon M\to \mathcal{D}^1_A$

Theorems & Definitions (44)

  • Example 1.1: Lie groups
  • Example 1.2: Riemannian manifolds
  • Example 1.3: $G$-structures with connection
  • Example 1.4: Space forms
  • Example 1.5: Metrics of Hessian type
  • Example 1.6: Extremal Kähler surfaces
  • Example 1.7: Surfaces with $|\nabla K| = 1$
  • Remark 1.8
  • Example 2.1: Exterior derivative
  • Example 2.2: Lie algebras
  • ...and 34 more