Borel subalgebra

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

If the Lie algebra $𝔤$ 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 $𝔤=𝔤𝔩(V)$ 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 $𝔤$ amounts to specify a flag of V; given a flag $V=V_0\supset V_1\supset \cdots \supset V_n=0$, the subspace $𝔟=\{x\in 𝔤\mid x(V_i)\subset V_i,1\le i\le n\}$ 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 $𝔤$ be a complex semisimple Lie algebra, $𝔥$ a Cartan subalgebra and R the root system associated to them. Choosing a base of R gives the notion of positive roots. Then $𝔤$ has the decomposition $𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+}$ where ${𝔫}^{\pm }=\sum _{\alpha >0}{𝔤}_{\pm \alpha }$. Then $𝔟=𝔥\oplus {𝔫}^{+}$ is the Borel subalgebra relative to the above setup. (It is solvable since the derived algebra $[𝔟,𝔟]$ is nilpotent. It is maximal solvable by a theorem of Borel–Morozov on the conjugacy of solvable subalgebras.)

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

See also

• Borel subgroup
• Parabolic Lie algebra

References

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