A category of locally convex Lie algebroids

Abstract

We study first-order locally convex Lie algebroids in the setting of Bastiani calculus. The first-order condition is automatic in finite rank, but is an additional regularity hypothesis for general locally convex vector bundles. We define sheaves of scalar-valued and vector-valued Lie algebroid forms as fiberwise continuous alternating maps with smooth local representatives. Morphisms are defined by requiring the induced pullback on inverse-image sheaves of scalar-valued forms to commute with the Lie algebroid differentials. We prove that these algebroids form a category, study representations and cohomology sheaves, and show that the Lie algebroids and morphisms of locally convex Lie groupoids belong to this category. We also give two applications. For a compact manifold K, the current algebroid C^∞(K,A)→ C^∞(K,M) of a first-order Banach Lie algebroid A→ M is a first-order Fréchet Lie algebroid, provided M admits local addition. Finally, we prove a Banach Lie II theorem: a morphism from the Lie algebroid of a locally convex Lie groupoid to that of a Banach-Lie groupoid, both with Hausdorff source fibers, integrates uniquely to a groupoid morphism when the source groupoid is source-connected and source-simply connected.

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BibTeX RIS

BibTeX

@misc{shaltut2026category,
  title = {{A category of locally convex Lie algebroids}},
  author = {Ahmed Gamal Shaltut},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.07672},
  eprint = {2607.07672},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.07672},
}

RIS

TY  - UNPB
TI  - A category of locally convex Lie algebroids
AU  - Ahmed Gamal Shaltut
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2607.07672
UR  - https://arxiv.org/abs/2607.07672
AB  - We study first-order locally convex Lie algebroids in the setting of Bastiani calculus. The first-order condition is automatic in finite rank, but is an additional regularity hypothesis for general locally convex vector bundles. We define sheaves of scalar-valued and vector-valued Lie algebroid forms as fiberwise continuous alternating maps with smooth local representatives. Morphisms are defined by requiring the induced pullback on inverse-image sheaves of scalar-valued forms to commute with the Lie algebroid differentials. We prove that these algebroids form a category, study representations and cohomology sheaves, and show that the Lie algebroids and morphisms of locally convex Lie groupoids belong to this category. We also give two applications. For a compact manifold K, the current algebroid C^∞(K,A)→ C^∞(K,M) of a first-order Banach Lie algebroid A→ M is a first-order Fréchet Lie algebroid, provided M admits local addition. Finally, we prove a Banach Lie II theorem: a morphism from the Lie algebroid of a locally convex Lie groupoid to that of a Banach-Lie groupoid, both with Hausdorff source fibers, integrates uniquely to a groupoid morphism when the source groupoid is source-connected and source-simply connected.
ER  -

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