Idempotent matrix: Difference between revisions

Revision as of 20:10, 24 February 2014

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-In [[mathematics]], the '''nilpotent cone''' <math>\mathcal{N}</math> of a finite-dimensional [[semisimple Lie algebra]] <math>\mathfrak{g}</math> is the set of elements that act nilpotently in all [[representation of Lie algebras|representations]] of  <math>\mathfrak{g}.</math> In other words,
-:<math> \mathcal{N}=\{ a\in \mathfrak{g}: \rho(a) \mbox{ is nilpotent for all representations } \rho:\mathfrak{g}\to \operatorname{End}(V)\}. </math>
-The nilpotent cone is an [[algebraic variety|irreducible subvariety]] of  <math>\mathfrak{g}</math> (considered as a <math>k</math>-[[vector space]]).
-==Example==
-The nilpotent cone of <math>\operatorname{sl}_2</math>, the Lie algebra of 2&times;2 [[matrix (mathematics)|matrices]] with vanishing [[trace (linear algebra)|trace]], is the variety of all 2&times;2 traceless matrices with [[rank (linear algebra)|rank]] less than or equal to <math>1.</math>
-{{PlanetMath attribution|id=4748|title=Nilpotent cone}}
-[[Category:Lie algebras]]
-{{algebra-stub}}

Idempotent matrix: Difference between revisions

Revision as of 20:10, 24 February 2014

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