Parabolic Lie algebra: Difference between revisions

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In algebra, a parabolic Lie algebra ${\mathfrak {p}}$ is a subalgebra of a semisimple Lie algebra ${\mathfrak {g}}$ satisfying one of the following two conditions:

${\mathfrak {p}}$ contains a maximal solvable subalgebra (a Borel subalgebra) of ${\mathfrak {g}}$ ;
the Killing perp of ${\mathfrak {p}}$ in ${\mathfrak {g}}$ is the nilradical of ${\mathfrak {p}}$ .

These conditions are equivalent over an algebraically closed field of characteristic zero, such as the complex numbers. If the field $\mathbb {F}$ is not algebraically closed, then the first condition is replaced by the assumption that

${\mathfrak {p}}\otimes _{\mathbb {F} }{\overline {\mathbb {F} }}$ contains a Borel subalgebra of ${\mathfrak {g}}\otimes _{\mathbb {F} }{\overline {\mathbb {F} }}$

where ${\overline {\mathbb {F} }}$ is the algebraic closure of $\mathbb {F}$ .

References

Robert J. Baston and Michael G. Eastwood, The Penrose Transform: its Interaction with Representation Theory, Oxford University Press, 1989.
William Fulton and Joe Harris (1991), Representation theory. A first course, Readings in Mathematics 129, Springer-Verlag.
Alexander Grothendieck (1957), Sur la classification des fibrés holomorphes sur la sphère de Riemann, Amer. J. Math. 79, 121–138.
J. Humphreys (1972). Linear Algebraic Groups. New York: Springer. ISBN 0-387-90108-6.

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-A '''parabolic Lie algebra''' '''p''' is a subalgebra of a [[semisimple Lie algebra]] '''g''' satisfying one of the following two equivalent conditions:
+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:
-* '''p''' contains a maximal [[solvable Lie algebra|solvable]] subalgebra (a [[Borel subalgebra]]) of '''g''';
+* <math>\mathfrak p</math> contains a maximal [[solvable Lie algebra|solvable]] subalgebra (a [[Borel subalgebra]]) of <math>\mathfrak g</math>;
+* the [[Killing form|Killing perp]] of <math>\mathfrak p</math> in <math>\mathfrak g</math> is the [[nilradical]] of <math>\mathfrak p</math>.
-* the [[nilradical]] of '''p''' is its [[Killing form|Killing perp]] in '''g'''.
+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
+* <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>
+where <math>\overline{\mathbb F}</math> is the [[algebraic closure]] of <math>\mathbb F</math>.
 ==See also==