Borel subalgebra

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

If the Lie algebra $\mathfrak$ 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 $\mathfrak g = \mathfrak(V)$ 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 = \$ 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 $\mathfrak n^ = \sum_ \mathfrak_$. 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/math> 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 $\mathfrak^+$. It is the same thing as a $\mathfrak b$-weight vector (Proof: if $h \in \mathfrak h$ and $e \in \mathfrak^+$ with $$, e= 2e and if $\mathfrak \cdot v$ is a line, then $0 =$, e\cdot v = 2 e \cdot v.)

See also

*Borel subgroup
* Parabolic Lie algebra

References

*.
*.
*.

{{algebra-stub

Representation theory
Lie algebras