In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra ${\displaystyle {\mathfrak {g))}$ is a maximal solvable subalgebra.^[1] The notion is named after Armand Borel.

If the Lie algebra ${\displaystyle {\mathfrak {g))}$ 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 ${\displaystyle {\mathfrak {g))={\mathfrak {gl))(V)}$ 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 ${\displaystyle {\mathfrak {g))}$ amounts to specify a flag of V; given a flag ${\displaystyle V=V_{0}\supset V_{1}\supset \cdots \supset V_{n}=0}$, the subspace ${\displaystyle {\mathfrak {b))=\{x\in {\mathfrak {g))\mid x(V_{i})\subset V_{i},1\leq i\leq n\))$ 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.

Borel subalgebra relative to a base of a root system

Let ${\displaystyle {\mathfrak {g))}$ be a complex semisimple Lie algebra, ${\displaystyle {\mathfrak {h))}$ a Cartan subalgebra and R the root system associated to them. Choosing a base of R gives the notion of positive roots. Then ${\displaystyle {\mathfrak {g))}$ has the decomposition ${\displaystyle {\mathfrak {g))={\mathfrak {n))^{-}\oplus {\mathfrak {h))\oplus {\mathfrak {n))^{+))$ where ${\displaystyle {\mathfrak {n))^{\pm }=\sum _{\alpha >0}{\mathfrak {g))_{\pm \alpha ))$. Then ${\displaystyle {\mathfrak {b))={\mathfrak {h))\oplus {\mathfrak {n))^{+))$ is the Borel subalgebra relative to the above setup.^[3] (It is solvable since the derived algebra ${\displaystyle [{\mathfrak {b)),{\mathfrak {b))]}$ is nilpotent. It is maximal solvable by a theorem of Borel–Morozov on the conjugacy of solvable subalgebras.^[4])

Given a ${\displaystyle {\mathfrak {g))}$-module V, a primitive element of V is a (nonzero) vector that (1) is a weight vector for ${\displaystyle {\mathfrak {h))}$ and that (2) is annihilated by ${\displaystyle {\mathfrak {n))^{+))$. It is the same thing as a ${\displaystyle {\mathfrak {b))}$-weight vector (Proof: if ${\displaystyle h\in {\mathfrak {h))}$ and ${\displaystyle e\in {\mathfrak {n))^{+))$ with ${\displaystyle [h,e]=2e}$ and if ${\displaystyle {\mathfrak {b))\cdot v}$ is a line, then ${\displaystyle 0=[h,e]\cdot v=2e\cdot v}$.)