Parabolic Lie algebra
Parabolic Lie algebras
In algebra, a parabolic Lie algebra p is a subalgebra of a semisimple Lie algebra g that can be described in two equivalent ways (over an algebraically closed field of characteristic zero, such as the complex numbers). If the ground field F is not algebraically closed, one uses the algebraic closure in place of one of these conditions.
In the gl_n setting: a parabolic subalgebra is the set of all n×n matrices that preserve a chain of subspaces 0 ⊆ V1 ⊆ V2 ⊆ ... ⊆ Vk ⊆ F^n. If the chain has all steps (a complete flag), the subalgebra is a Borel subalgebra. If you only require preserving a single subspace F^k ⊆ F^n, you get a maximal parabolic subalgebra; different choices of the k-dimensional subspace correspond to the Grassmannian Gr(k,n).
In general for a complex simple Lie algebra g, parabolic subalgebras correspond to subsets of the simple roots (or equivalently, to subsets of the nodes of the Dynkin diagram).
This page was last edited on 2 February 2026, at 03:09 (CET).