Parabolic Lie algebra

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In algebra, a parabolic Lie algebra 𝔭 is a subalgebra of a semisimple Lie algebra 𝔤 satisfying one of the following two conditions:

* 𝔭 contains a maximal solvable subalgebra (a Borel subalgebra) of 𝔤;
* the orthogonal complement with respect to the Killing form of 𝔭 in 𝔤 is isomorphic to the nilradical of 𝔭.

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

* 𝔭⊗𝔽𝔽‾ contains a Borel subalgebra of 𝔤⊗𝔽𝔽‾

where 𝔽‾ is the algebraic closure of 𝔽.

## Examples

For the general linear Lie algebra 𝔤=𝔤𝔩n(𝔽), a parabolic subalgebra is the stabilizer of a partial flag of 𝔽n, i.e. a sequence of nested linear subspaces. For a complete flag, the stabilizer gives a Borel subalgebra. For a single linear subspace 𝔽k⊂𝔽n, one gets a maximal parabolic subalgebra 𝔭, and the space of possible choices is the Grassmannian Gr(k,n).

In general, for a complex simple Lie algebra 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), 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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