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 $\mathfrak{𝔤}=\mathfrak{𝔤}\mathfrak{𝔩}(V)$ 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 $0=[h,e]\cdot v=2e\cdot v$.)

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 .