Parabolic subalgebra

Summary

A parabolic subalgebra of a finite-dimensional Lie algebra over an algebraically closed field of characteristic $0$ is a subalgebra that contains a Borel subalgebra. In $\mathfrak{gl}_n(\C)$ the parabolic subalgebras containing the upper triangular matrices are the block upper triangular matrices. They are the Lie algebras of parabolic subgroups, and they are used to build induced representations and generalized flag varieties.