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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 $$\mathbb F$$ .

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

Mathematics Encyclopedia