For a smooth left Lie group action, define the fundamental vector field by . With the Lie bracket of vector fields , this map is an anti-homomorphism. The map is a Lie algebra representation. A right action with the positive exponential has the opposite homomorphism convention, so specifying action side and bracket convention prevents a sign ambiguity.
Articles by others on the same topic
In differential geometry, the term "fundamental vector field" often refers to a particular type of vector field associated with a group action on a manifold. Specifically, when a Lie group acts on a differentiable manifold, each element of the Lie algebra of the group can be associated with a vector field on the manifold known as a fundamental vector field. ### Definition Let \( G \) be a Lie group acting smoothly on a differentiable manifold \( M \).