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 $\mathbb F$ .

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. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
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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Parabolic Lie algebra

See also

Bibliography