Parabolic Subalgebra

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
* $\mathfrak p\otimes_\overline$ contains a Borel subalgebra of $\mathfrak g\otimes_\overline$
where $\overline$ is the algebraic closure of $\mathbb F$.

See also

* Generalized flag variety

Bibliography

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Lie algebras

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