Parabolic Lie algebra

Template:Inline In algebra, a parabolic Lie algebra $𝔭$ is a subalgebra of a semisimple Lie algebra $𝔤$ satisfying one of the following two conditions:

• $𝔭$ contains a maximal solvable subalgebra (a Borel subalgebra) of $𝔤$;
• the orthogonal complement with respect to the Killing form of $𝔭$ in $𝔤$ is the nilradical of $𝔭$.

These conditions are equivalent over an algebraically closed field of characteristic zero, such as the complex numbers. If the field $𝔽$ is not algebraically closed, then the first condition is replaced by the assumption that

• $𝔭{\otimes }_{𝔽}\overline{𝔽}$ contains a Borel subalgebra of $𝔤{\otimes }_{𝔽}\overline{𝔽}$

where $\overline{𝔽}$ is the algebraic closure of $𝔽$.

Examples

For the general linear Lie algebra $𝔤={𝔤𝔩}_n(𝔽)$, a parabolic subalgebra is the stabilizer of a partial flag of ${𝔽}^n$, i.e. a sequence of nested linear subspaces. For a complete flag, the stabilizer gives a Borel subalgebra. For a single linear subspace ${𝔽}^k\subset {𝔽}^n$, one gets a maximal parabolic subalgebra $𝔭$, and the space of possible choices is the Grassmannian $\mathrm{G}\mathrm{r}(k,n)$.

In general, for a complex simple Lie algebra $𝔤$, parabolic subalgebras are in bijection with subsets of simple roots, i.e. subsets of the nodes of the Dynkin diagram.

See also

• Generalized flag variety
• Parabolic subgroup of a reflection group

Bibliography

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