Abstract
This chapter discusses the nilpotent and solvable Lie algebras. If A is nilpotent and V1 is invariant under A, then Ā is also nilpotent. From nilpotency of A and by induction, it can be proved that all eigenvalues of a nilpotent linear transformation are zero. Conversely, if all eigenvalues of a linear transformation are zero, it must be nilpotent. A Lie algebra g is nilpotent if for any X ∈ g, ad X is nilpotent. According to Lie's theorem, any solvable linear Lie algebra, in particular, any nilpotent linear Lie algebra, has at least one weight.
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