Borel subalgebra

In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] 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 [math]\displaystyle{ \mathfrak g = \mathfrak{gl}(V) }[/math] 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 [math]\displaystyle{ \mathfrak g }[/math] amounts to specify a flag of V; given a flag [math]\displaystyle{ V = V_0 \supset V_1 \supset \cdots \supset V_n = 0 }[/math], the subspace [math]\displaystyle{ \mathfrak b = \{ x \in \mathfrak g \mid x(V_i) \subset V_i, 1 \le i \le n \} }[/math] 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 [math]\displaystyle{ \mathfrak g }[/math] be a complex semisimple Lie algebra, [math]\displaystyle{ \mathfrak h }[/math] a Cartan subalgebra and R the root system associated to them. Choosing a base of R gives the notion of positive roots. Then [math]\displaystyle{ \mathfrak g }[/math] has the decomposition [math]\displaystyle{ \mathfrak g = \mathfrak n^- \oplus \mathfrak h \oplus \mathfrak n^+ }[/math] where [math]\displaystyle{ \mathfrak n^{\pm} = \sum_{\alpha \gt 0} \mathfrak{g}_{\pm \alpha} }[/math]. Then [math]\displaystyle{ \mathfrak b = \mathfrak h \oplus \mathfrak n^+ }[/math] is the Borel subalgebra relative to the above setup. (It is solvable since the derived algebra [math]\displaystyle{ [\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 [math]\displaystyle{ \mathfrak g }[/math]-module V, a primitive element of V is a (nonzero) vector that (1) is a weight vector for [math]\displaystyle{ \mathfrak h }[/math] and that (2) is annihilated by [math]\displaystyle{ \mathfrak{n}^+ }[/math]. It is the same thing as a [math]\displaystyle{ \mathfrak b }[/math]-weight vector (Proof: if [math]\displaystyle{ h \in \mathfrak h }[/math] and [math]\displaystyle{ e \in \mathfrak{n}^+ }[/math] with [math]\displaystyle{ [h, e] = 2e }[/math] and if [math]\displaystyle{ \mathfrak{b} \cdot v }[/math] is a line, then [math]\displaystyle{ 0 = [h, e] \cdot v = 2 e \cdot v }[/math].)

See also

• Borel subgroup
• Parabolic Lie algebra

References

• Chriss, Neil; Ginzburg, Victor (2009), Representation Theory and Complex Geometry, Springer, ISBN 978-0-8176-4938-8, https://books.google.com/books?id=OZlCAAAAQBAJ&pg=PP1 .
• Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Springer-Verlag, ISBN 978-0-387-90053-7, https://archive.org/details/introductiontoli00jame/page/83 .
• Serre, Jean-Pierre (2000) (in en), Algèbres de Lie semi-simples complexes, Springer, ISBN 978-3-540-67827-4, https://books.google.com/books?id=7AHsSUrooSsC&pg=PA3 .