Parabolic Lie algebra

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

contains a maximal solvable subalgebra of ;
the orthogonal complement with respect to the Killing form of in is isomorphic to the nilradical of.

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

contains a Borel subalgebra of

where is the algebraic closure of.

Examples

For the general linear Lie algebra, a parabolic subalgebra is the stabilizer of a partial flag of, i.e. a sequence of nested linear subspaces. For a complete flag, the stabilizer gives a Borel subalgebra. For a single linear subspace, one gets a maximal parabolic subalgebra, and the space of possible choices is the Grassmannian.
In general, for a complex simple Lie algebra, parabolic subalgebras are in bijection with subsets of simple roots, i.e. subsets of the nodes of the Dynkin diagram.