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 orthogonal complement with respect to the Killing form 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

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

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

Examples

For the general linear Lie algebra ${\displaystyle {\mathfrak {g}}={\mathfrak {gl}}_{n}(\mathbb {F} )}$, a parabolic subalgebra is the stabilizer of a partial flag of ${\displaystyle \mathbb {F} ^{n}}$, i.e. a sequence of nested linear subspaces. For a complete flag, the stabilizer gives a Borel subalgebra. For a single linear subspace ${\displaystyle \mathbb {F} ^{k}\subset \mathbb {F} ^{n}}$, one gets a maximal parabolic subalgebra ${\displaystyle {\mathfrak {p}}}$, and the space of possible choices is the Grassmannian ${\displaystyle \mathrm {Gr} (k,n)}$.

In general, for a complex simple Lie algebra ${\displaystyle {\mathfrak {g}}}$, parabolic subalgebras are in bijection with subsets of simple roots, i.e. subsets of the nodes of the Dynkin diagram.

See also

• Generalized flag variety
• Parabolic subgroup of a reflection group

Bibliography

• Baston, Robert J.; Eastwood, Michael G. (2016) [1989], The Penrose Transform: its Interaction with Representation Theory, Dover, ISBN 9780486816623
• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 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, Springer, ISBN 978-0-387-90108-4