Lie Algebras of vector fields on convenient manifolds

TL;DR

The paper explores different definitions of vector fields on convenient manifolds that form Lie algebras, establishing their equivalence to the standard finite-dimensional notion.

Contribution

It compares old and new definitions of vector fields on convenient manifolds, proving their equivalence to classical finite-dimensional definitions.

Findings

Various definitions of vector fields on convenient manifolds are equivalent to the standard notion.

The paper clarifies the structure of Lie algebras formed by these vector fields.

Connections between different approaches to vector fields are established.

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.