Borel subalgebra

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

If the Lie algebra $$\mathfrak{g}$$ 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 $$\mathfrak g = \mathfrak{gl}(V)$$ 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 $$\mathfrak g$$ amounts to specify a flag of V; given a flag $$V = V_0 \supset V_1 \supset \cdots \supset V_n = 0$$, the subspace $$\mathfrak b = \{ x \in \mathfrak g \mid x(V_i) \subset V_i, 1 \le i \le n \}$$ 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.

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

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