Parabolic Lie algebra

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:

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

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

$\mathfrak p\otimes_{\mathbb F}\overline{\mathbb F}$ contains a Borel subalgebra of $\mathfrak g\otimes_{\mathbb F}\overline{\mathbb F}$

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

Examples

For the general linear Lie algebra $\mathfrak{g}=\mathfrak{gl}_n(\mathbb F)$ , 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 $\mathbb F^k\subset \mathbb F^n$ , one gets a maximal parabolic subalgebra $\mathfrak p$ , and the space of possible choices is the Grassmannian $\mathrm{Gr}(k,n)$ .

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.

Bibliography

{{citation|first1=Robert J.|last1=Baston|first2=Michael G.|last2=Eastwood|authorlink2=Michael Eastwood|title=The Penrose Transform: its Interaction with Representation Theory|publisher=Dover|orig-year=1989 |year=2016 |url=https://books.google.com/books?id=MEVmDQAAQBAJ |isbn=9780486816623}}
{{Fulton-Harris}}
{{citation|doi=10.2307/2372388|first=Alexander|last=Grothendieck|author-link=Alexander Grothendieck|year=1957|title=Sur la classification des fibrés holomorphes sur la sphère de Riemann|journal=Amer. J. Math.|volume=79|issue=1|pages=121–138|jstor=2372388}}.
{{citation | first=J.|last=Humphreys | title=Linear Algebraic Groups | publisher=Springer | year=1972 | isbn=978-0-387-90108-4}}

Category:Lie algebras

Parabolic Lie algebra

Examples

See also

Bibliography