Borel subalgebra

In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra ${\displaystyle {\mathfrak {g}}}$is a maximal solvable subalgebra. 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, 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. (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.)

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

• Borel subgroup
• Parabolic Lie algebra