Parabolic Lie algebra

In algebra, a parabolic Lie algebra [math]\displaystyle{ \mathfrak p }[/math] is a subalgebra of a semisimple Lie algebra [math]\displaystyle{ \mathfrak g }[/math] satisfying one of the following two conditions:

• [math]\displaystyle{ \mathfrak p }[/math] contains a maximal solvable subalgebra (a Borel subalgebra) of [math]\displaystyle{ \mathfrak g }[/math];
• the Killing perp of [math]\displaystyle{ \mathfrak p }[/math] in [math]\displaystyle{ \mathfrak g }[/math] is the nilradical of [math]\displaystyle{ \mathfrak p }[/math].

These conditions are equivalent over an algebraically closed field of characteristic zero, such as the complex numbers. If the field [math]\displaystyle{ \mathbb F }[/math] is not algebraically closed, then the first condition is replaced by the assumption that

• [math]\displaystyle{ \mathfrak p\otimes_{\mathbb F}\overline{\mathbb F} }[/math] contains a Borel subalgebra of [math]\displaystyle{ \mathfrak g\otimes_{\mathbb F}\overline{\mathbb F} }[/math]

where [math]\displaystyle{ \overline{\mathbb F} }[/math] is the algebraic closure of [math]\displaystyle{ \mathbb F }[/math].

See also

• Generalized flag variety
• Parabolic subgroup of a reflection group

Bibliography

• Baston, Robert J.; Eastwood, Michael G. (2016), The Penrose Transform: its Interaction with Representation Theory, Dover, ISBN 9780486816623, https://books.google.com/books?id=MEVmDQAAQBAJ 
• Fulton, William; Harris, Joe (1991) (in en-gb). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. OCLC 246650103. https://link.springer.com/10.1007/978-1-4612-0979-9. 
• 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 .
• Humphreys, J. (1972), Linear Algebraic Groups, Springer, ISBN 978-0-387-90108-4