Borel subalgebra

In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra

is a maximal solvable subalgebra.

[1] The notion is named after Armand Borel.

If the Lie algebra

is the Lie algebra of a complex Lie group, then a Borel subalgebra is the Lie algebra of a Borel subgroup.

{\displaystyle {\mathfrak {g}}={\mathfrak {gl}}(V)}

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

amounts to specify a flag of V; given a flag

, the subspace

is a Borel subalgebra,[2] 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. Let

be a complex semisimple Lie algebra,

a Cartan subalgebra and R the root system associated to them.

Choosing a base of R gives the notion of positive roots.

has the decomposition

is the Borel subalgebra relative to the above setup.

[3] (It is solvable since the derived algebra

is nilpotent.

It is maximal solvable by a theorem of Borel–Morozov on the conjugacy of solvable subalgebras.

-module V, a primitive element of V is a (nonzero) vector that (1) is a weight vector for

-weight vector (Proof: if

is a line, then

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