Borel subalgebra explained

ak{g} is a maximal solvable subalgebra. The notion is named after Armand Borel.

If the Lie algebra

ak{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

ak g = ak{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 ak g amounts to specify a flag of V; given a flag V = V _{0 \supset} V _{1} \supset … \supset V _{n} = 0 , the subspace ak b = \{ x \in ak 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

ak g be a complex semisimple Lie algebra, ak h a Cartan subalgebra and R the root system associated to them. Choosing a base of R gives the notion of positive roots. Then ak g has the decomposition ak g = ak n ^{-} ⊕ ak h ⊕ ak n ^{+} where ak n ^{\pm} = \sum _{\alpha} ak{g} _{\pm} . Then ak b = ak h ⊕ ak n ^{+} is the Borel subalgebra relative to the above setup. (It is solvable since the derived algebra [ ak b, ak b] is nilpotent. It is maximal solvable by a theorem of Borel–Morozov on the conjugacy of solvable subalgebras.)

Given a

ak g -module V, a primitive element of V is a (nonzero) vector that (1) is a weight vector for ak h and that (2) is annihilated by ak{n} ^{+} . It is the same thing as a ak b -weight vector (Proof: if h \in ak h and e \in ak{n} ^{+} with [h, e] = 2 e and if ak{b} ⋅ v is a line, then 0 = [h, e] ⋅ v = 2 e ⋅ v .)

See also

• Borel subgroup
• Parabolic Lie algebra

References

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