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| In [[algebra]], a '''parabolic Lie algebra''' <math>\mathfrak p</math> is a subalgebra of a [[semisimple Lie algebra]] <math>\mathfrak g</math> satisfying one of the following two conditions:
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| * <math>\mathfrak p</math> contains a maximal [[solvable Lie algebra|solvable]] subalgebra (a [[Borel subalgebra]]) of <math>\mathfrak g</math>;
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| * the [[Killing form|Killing perp]] of <math>\mathfrak p</math> in <math>\mathfrak g</math> is the [[Nilradical of a Lie algebra|nilradical]] of <math>\mathfrak p</math>.
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| These conditions are equivalent over an [[algebraically closed]] [[field (mathematics)|field]] of [[characteristic zero]], such as the [[complex numbers]]. If the field <math>\mathbb F</math> is not algebraically closed, then the first condition is replaced by the assumption that
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| * <math>\mathfrak p\otimes_{\mathbb F}\overline{\mathbb F}</math> contains a Borel subalgebra of <math> \mathfrak g\otimes_{\mathbb F}\overline{\mathbb F}</math>
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| where <math>\overline{\mathbb F}</math> is the [[algebraic closure]] of <math>\mathbb F</math>.
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| ==See also==
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| * [[Generalized flag variety]]
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| ==Bibliography==
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| * {{citation|first1=Robert J.|last1=Baston|first2=Michael G.|last2=Eastwood|authorlink2=Michael Eastwood|title=The Penrose Transform: its Interaction with Representation Theory|publisher=Oxford University Press|year=1989}}.
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| * {{Fulton-Harris}}
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| * {{citation|doi=10.2307/2372388|first=Alexander|last=Grothendieck|authorlink=Alexander Grothendieck|year=1957|title=Sur la classification des fibrés holomorphes sur la sphère de Riemann|journal=Amer. J. Math.|volume=79|issue=1|pages=121–138|jstor=2372388}}.
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| * {{citation | first=J.|last=Humphreys | title=Linear Algebraic Groups | location=New York | publisher=Springer | year=1972 | id=ISBN 0-387-90108-6}}
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| [[Category:Lie algebras]]
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| {{algebra-stub}}
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Latest revision as of 00:07, 11 January 2015
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