Parabolic Lie algebra

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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 isomorphic to 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 ${\displaystyle \mathbb{𝔽}}$.

For the general linear Lie algebra ${\displaystyle \mathfrak{𝔤}={\mathfrak{𝔤}\mathfrak{𝔩}}_n(\mathbb{𝔽})}$, 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 ${\displaystyle \mathrm{G}\mathrm{r}(k,n)}$.

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.

• Generalized flag variety
• Parabolic subgroup of a reflection group

• 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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