Borel subalgebra

In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra formula_1 is a maximal solvable subalgebra. The notion is named after Armand Borel.
If the Lie algebra formula_1 is the Lie algebra of a complex Lie group, then a Borel subalgebra is the Lie algebra of a Borel subgroup.
Borel subalgebra associated to a flag.
Let formula_3 be the Lie algebra of the endomorphisms of a finite-dimensional vector space "V" over the complex numbers. Then to specify a Borel subalgebra of formula_4 amounts to specify a flag of "V"; given a flag formula_5, the subspace formula_6 is a Borel subalgebra, and conversely, each Borel subalgebra is of that form by Lie's theorem. Hence, the Borel subalgebras are classified by the flag variety of "V".
Borel subalgebra relative to a base of a root system.
Let formula_4 be a complex semisimple Lie algebra, formula_8 a Cartan subalgebra and "R" the root system associated to them. Choosing a base of "R" gives the notion of positive roots. Then formula_4 has the decomposition formula_10 where formula_11. Then formula_12 is the Borel subalgebra relative to the above setup. (It is solvable since the derived algebra formula_13 is nilpotent. It is maximal solvable by a theorem of Borel–Morozov on the conjugacy of solvable subalgebras.)
Given a formula_4-module "V", a primitive element of "V" is a (nonzero) vector that (1) is a weight vector for formula_8 and that (2) is annihilated by formula_16. It is the same thing as a formula_17-weight vector (Proof: if formula_18 and formula_19 with formula_20 and if formula_21 is a line, then formula_22.)