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Lie Algebras of vector fields on convenient manifolds

Arnold Neumaier, Phillip Josef Bachler

TL;DR

The paper addresses how to formulate Lie algebras of vector fields on infinite-dimensional, convenient manifolds by unifying several definitions with a coherent calculus. It develops an array of concepts—infinitesimal motions, algebraic, accessible, operational, and kinematic vector fields—together with a formalism for asymptotic expansions (IS and o_f) and group motions, to produce Lie-algebra structures via derivations and commutators. The work provides a rigorous foundation for vector-field Lie algebras in infinite dimensions, including a Lie-derivative framework, tensor calculus, and differential-operator machinery, with explicit embedding relations and calculus rules. This framework supports applications to coherent manifolds and broader geometric analysis, offering tools for PDEs, quantum field theory, and related areas where infinite-dimensional differential geometry is essential.

Abstract

We discuss various old and new definitions of the notion of a vector field on a convenient manifold that can be proved to give rise to Lie algebras, and are in finite dimensions equivalent to the standard notion of a vector field.

Lie Algebras of vector fields on convenient manifolds

TL;DR

The paper addresses how to formulate Lie algebras of vector fields on infinite-dimensional, convenient manifolds by unifying several definitions with a coherent calculus. It develops an array of concepts—infinitesimal motions, algebraic, accessible, operational, and kinematic vector fields—together with a formalism for asymptotic expansions (IS and o_f) and group motions, to produce Lie-algebra structures via derivations and commutators. The work provides a rigorous foundation for vector-field Lie algebras in infinite dimensions, including a Lie-derivative framework, tensor calculus, and differential-operator machinery, with explicit embedding relations and calculus rules. This framework supports applications to coherent manifolds and broader geometric analysis, offering tools for PDEs, quantum field theory, and related areas where infinite-dimensional differential geometry is essential.

Abstract

We discuss various old and new definitions of the notion of a vector field on a convenient manifold that can be proved to give rise to Lie algebras, and are in finite dimensions equivalent to the standard notion of a vector field.

Paper Structure

This paper contains 26 sections, 29 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lie algebras of infinitesimal motions
  3. Infinitesimally smooth maps
  4. The fractional Landau symbol ofmTE
  5. Motions on groups
  6. Lie algebras
  7. Lie algebras of vector fields
  8. Algebraic vector fields
  9. Accessible vector fields
  10. Operational vector fields
  11. Kinematic vector fields
  12. Summary chart
  13. Further concepts
  14. The Lie derivative
  15. Tensor fields
  16. ...and 11 more sections

Key Result

Theorem 2.1

Neumaier NeuInt] Let $F,G \in \mathop{\mathrm{IS}}\nolimits(T,\mathbb{E})$ with $T$ a $0$-set and $\mathbb{E}$ an associative algebra.

Figures (2)

  • Figure 1: Embedding relations for Lie algebras of vector fields
  • Figure 2: Embedding relations for Lie algebras of vector fields

Theorems & Definitions (29)

  • Theorem 2.1: cf. Neumaierauthor
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4: cf. Neumaier & Westraauthor
  • Proposition 2.5
  • Theorem 2.6: cf. Neumaier & Westraauthor
  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • ...and 19 more