In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak{g} is a maximal solvable subalgebra.[1] 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 \mid x(V_i) \subset V_i, 1 \le i \le n \} is a Borel subalgebra,[2] 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.[3] (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.[4])
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 = 2 e \cdot v.)
See also
References
- ^ Humphreys, pp. Ch XVI, § 3.
- ^ Serre 2000, pp. Ch I, § 6.
- ^ Serre 2000, pp. Ch VI, § 3.
- ^ Serre 2000, pp. Ch. VI, § 3. Theorem 5.
- 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.