Parabolic Lie algebra

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

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

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

• ${\displaystyle \mathfrak{p}{\otimes }_{\mathbb{F}}\bar{\mathbb{F}}}$ contains a Borel subalgebra of ${\displaystyle \mathfrak{g}{\otimes }_{\mathbb{F}}\bar{\mathbb{F}}}$

where ${\displaystyle \bar{\mathbb{F}}}$ is the algebraic closure of ${\displaystyle \mathbb{F}}$.

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 

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