Parabolic Lie algebra

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

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

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

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

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