Parabolic Lie algebra
- contains a maximal solvable subalgebra (a Borel subalgebra) of ;
- the Killing perp of in is the nilradical of .
These conditions are equivalent over an algebraically closed field of characteristic zero, such as the complex numbers. If the field is not algebraically closed, then the first condition is replaced by the assumption that
- contains a Borel subalgebra of
where is the algebraic closure of .
- 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.
- 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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