Parabolic Lie algebra explained
In algebra, a parabolic Lie algebra
is a subalgebra of a
semisimple Lie algebra
satisfying one of the following two conditions:
contains a maximal
solvable subalgebra (a
Borel subalgebra) of
;
- the orthogonal complement with respect to the Killing form of
in
is the
nilradical of
.These conditions are equivalent over an algebraically closed
field of
characteristic zero, such as the complex numbers. If the field
is not algebraically closed, then the first condition is replaced by the assumption that
contains a Borel subalgebra of
where
is the
algebraic closure of
.
Examples
For the general linear Lie algebra
, a parabolic subalgebra is the stabilizer of a partial
flag of
, i.e. a sequence of nested linear subspaces. For a complete flag, the stabilizer gives a Borel subalgebra. For a single linear subspace
, one gets a maximal parabolic subalgebra
, and the space of possible choices is the
Grassmannian
.
In general, for a complex simple Lie algebra
, parabolic subalgebras are in bijection with subsets of simple
roots, i.e. subsets of the nodes of the
Dynkin diagram.
See also
Bibliography
