In algebra, a parabolic Lie algebra p {\displaystyle {\mathfrak {p}}} is a subalgebra of a semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} satisfying one of the following two conditions:

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

  • p ⊗ F F ¯ {\displaystyle {\mathfrak {p}}\otimes _{\mathbb {F} }{\overline {\mathbb {F} }}} contains a Borel subalgebra of g ⊗ F F ¯ {\displaystyle {\mathfrak {g}}\otimes _{\mathbb {F} }{\overline {\mathbb {F} }}}

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

Examples

For the general linear Lie algebra g = g l n ( F ) {\displaystyle {\mathfrak {g}}={\mathfrak {gl}}_{n}(\mathbb {F} )}, a parabolic subalgebra is the stabilizer of a partial flag of F n {\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 F k ⊂ F n {\displaystyle \mathbb {F} ^{k}\subset \mathbb {F} ^{n}}, one gets a maximal parabolic subalgebra p {\displaystyle {\mathfrak {p}}}, and the space of possible choices is the Grassmannian G r ( k , n ) {\displaystyle \mathrm {Gr} (k,n)}.

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

See also

Bibliography