Definition
Let be a Lie group with corresponding Lie algebra . Furthermore, let be a smooth manifold endowed with a smooth action . Denote the map such that, called the orbit map of corresponding to . For, the fundamental vector field corresponding to is any of the following equivalent definitions:
where is the exterior derivative and is the zero vector in the vector space .
The map can then be shown to be a Lie algebra homomorphism.
Read more about this topic: Fundamental Vector Field
Famous quotes containing the word definition:
“Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
—The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)