Definition
An element X of a semisimple Lie algebra g is called nilpotent if its adjoint endomorphism
- ad X: g → g, ad X(Y) =
is nilpotent, that is, (ad X)n = 0 for large enough n. Equivalently, X is nilpotent if its characteristic polynomial pad X(t) is equal to tdim g.
A semisimple Lie group or algebraic group G acts on its Lie algebra via the adjoint representation, and the property of being nilpotent is invariant under this action. A nilpotent orbit is an orbit of the adjoint action such that any (equivalently, all) of its elements is (are) nilpotent.
Read more about this topic: Nilpotent Orbit
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