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 Killing perp 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
⊗
F
F
¯
contains a Borel subalgebra of
g
⊗
F
F
¯
where
F
¯
is the algebraic closure of
F
.
