In algebra, a parabolic Lie algebra \mathfrak p is a subalgebra of a semisimple Lie algebra \mathfrak g satisfying one of the following two conditions: These conditions are equivalent over an algebraically closed field of characteristic zero, such as the complex numbers. If the field \mathbb F is not algebraically closed, then the first condition is replaced by the assumption that where is the algebraic closure of \mathbb F.

Examples

For the general linear Lie algebra, a parabolic subalgebra is the stabilizer of a partial flag of \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, one gets a maximal parabolic subalgebra \mathfrak p, and the space of possible choices is the Grassmannian. In general, for a complex simple Lie algebra \mathfrak g, parabolic subalgebras are in bijection with subsets of simple roots, i.e. subsets of the nodes of the Dynkin diagram.