Parabolic Lie algebra

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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 Killing perp 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 .

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