Parabolic Lie algebra

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
* $mathfrak\; potimes\_\{mathbb\; F\}overline\{mathbb\; F\}$ contains a Borel subalgebra of $mathfrak\; gotimes\_\{mathbb\; F\}overline\{mathbb\; F\}$where $overline\{mathbb\; F\}$ is the algebraic closure of $mathbb\; F$.

ee also

* Generalized flag variety

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.
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