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
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