In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra g {\displaystyle {\mathfrak {g}}} is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra g {\displaystyle {\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 g = g l ( V ) {\displaystyle {\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 g {\displaystyle {\mathfrak {g}}} amounts to specify a flag of V; given a flag V = V 0 ⊃ V 1 ⊃ ⋯ ⊃ V n = 0 {\displaystyle V=V_{0}\supset V_{1}\supset \cdots \supset V_{n}=0} , the subspace b = { x ∈ g ∣ x ( V i ) ⊂ V i , 1 ≤ i ≤ n } {\displaystyle {\mathfrak {b}}=\{x\in {\mathfrak {g}}\mid 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 g {\displaystyle {\mathfrak {g}}} be a complex semisimple Lie algebra, h {\displaystyle {\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 g {\displaystyle {\mathfrak {g}}} has the decomposition g = n − ⊕ h ⊕ n + {\displaystyle {\mathfrak {g}}={\mathfrak {n}}^{-}\oplus {\mathfrak {h}}\oplus {\mathfrak {n}}^{+}} where n ± = ∑ α > 0 g ± α {\displaystyle {\mathfrak {n}}^{\pm }=\sum _{\alpha >0}{\mathfrak {g}}_{\pm \alpha }} . Then b = h ⊕ n + {\displaystyle {\mathfrak {b}}={\mathfrak {h}}\oplus {\mathfrak {n}}^{+}} is the Borel subalgebra relative to the above setup. (It is solvable since the derived algebra [ b , b ] {\displaystyle [{\mathfrak {b}},{\mathfrak {b}}]} is nilpotent. It is maximal solvable by a theorem of Borel–Morozov on the conjugacy of solvable subalgebras.) Given a g {\displaystyle {\mathfrak {g}}} -module V, a primitive element of V is a (nonzero) vector that (1) is a weight vector for h {\displaystyle {\mathfrak {h}}} and that (2) is annihilated by n + {\displaystyle {\mathfrak {n}}^{+}} . It is the same thing as a b {\displaystyle {\mathfrak {b}}} -weight vector (Proof: if h ∈ h {\displaystyle h\in {\mathfrak {h}}} and e ∈ n + {\displaystyle e\in {\mathfrak {n}}^{+}} with [ h , e ] = 2 e {\displaystyle [h,e]=2e} and if b ⋅ v {\displaystyle {\mathfrak {b}}\cdot v} is a line, then 0 = [ h , e ] ⋅ v = 2 e ⋅ v {\displaystyle 0=[h,e]\cdot v=2e\cdot v} .)
See also Borel subgroup Parabolic Lie algebra
References