is a subalgebra of a semisimple Lie algebra
satisfying one of the following two conditions: These conditions are equivalent over an algebraically closed field of characteristic zero, such as the complex numbers.
is not algebraically closed, then the first condition is replaced by the assumption that where
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.
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.