Properties
Let be a finite dimensional Lie algebra over a field of characteristic 0. The following are equivalent.
- (i) is solvable.
- (ii), the adjoint representation of, is solvable.
- (iii) There is a finite sequence of ideals of such that:
- where for all .
- (iv) is nilpotent.
Lie's Theorem states that if is a finite-dimensional vector space over an algebraically closed field of characteristic zero, and is a solvable linear Lie algebra over, then there exists a basis of relative to which the matrices of all elements of are upper triangular.
Read more about this topic: Solvable Lie Algebra
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)