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 ${\mathfrak {b}}=\{x\in {\mathfrak {g}}|x(V_{i})\subset V_{i},1\leq i\leq n\}$ 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) [1997], Representation Theory and Complex Geometry, Springer, ISBN 978-0-8176-4938-8.
• Humphreys, James E. (1972), Introduction to Lie Algebras and Representation Theory, Springer-Verlag, ISBN 978-0-387-90053-7.
• Serre, Jean-Pierre (2000), Algèbres de Lie semi-simples complexes [Complex Semisimple Lie Algebras], translated by Jones, G. A., Springer, ISBN 978-3-540-67827-4.