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

Examples

For the general linear Lie algebra ${\displaystyle {\mathfrak {g))={\mathfrak {gl))_{n}(\mathbb {F} )}$, 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 ${\displaystyle \mathrm {Gr} (k,n)}$.

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.