Borel subalgebra

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.

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.

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 ${\displaystyle 0=[h,e]\cdot v=2e\cdot v}$.)