In
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
, a parabolic Lie algebra
is a subalgebra of a
semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra is ...
satisfying one of the following two conditions:
*
contains a maximal
solvable subalgebra (a
Borel subalgebra In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable subalgebra. The notion is named after Armand Borel.
If the Lie algebra \mathfrak is the Lie algebra of a complex Lie group, ...
) of
;
* the
Killing perp of
in
is the
nilradical of
.
These conditions are equivalent over an
algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, becaus ...
field of
characteristic zero, such as the
complex numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
. 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
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ...
of
.
See also
*
Generalized flag variety
Bibliography
*
*
* .
*
Lie algebras
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