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In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important role in representation theory of real and complex semisimple Lie groups and semisimple Lie algebras. == 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 ''p''''ad X''(''t'') is equal to ''t''dim ''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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「nilpotent orbit」の詳細全文を読む スポンサード リンク
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