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

\( \mathfrak p\otimes_{\mathbb F}\overline{\mathbb F} contains a Borel subalgebra of \mathfrak g\otimes_{\mathbb F}\overline{\mathbb F} \)

where \( \overline{\mathbb F} \) is the algebraic closure of \( \mathbb F \) .
See also

Generalized flag variety

Bibliography

Baston, Robert J.; Eastwood, Michael G. (1989), The Penrose Transform: its Interaction with Representation Theory, Oxford University Press.
Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6
Grothendieck, Alexander (1957), "Sur la classification des fibrés holomorphes sur la sphère de Riemann", Amer. J. Math. 79 (1): 121–138, doi:10.2307/2372388, JSTOR 2372388.
Humphreys, J. (1972), Linear Algebraic Groups, New York: Springer, ISBN 0-387-90108-6

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