Английская Википедия:Borel subalgebra

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra <math>\mathfrak{g}</math> is a maximal solvable subalgebra. The notion is named after Armand Borel.

If the Lie algebra <math>\mathfrak{g}</math> 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 <math>\mathfrak g = \mathfrak{gl}(V)</math> 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 <math>\mathfrak g</math> amounts to specify a flag of V; given a flag <math>V = V_0 \supset V_1 \supset \cdots \supset V_n = 0</math>, the subspace <math>\mathfrak b = \{ x \in \mathfrak g \mid x(V_i) \subset V_i, 1 \le i \le n \}</math> 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 <math>\mathfrak g</math> be a complex semisimple Lie algebra, <math>\mathfrak h</math> a Cartan subalgebra and R the root system associated to them. Choosing a base of R gives the notion of positive roots. Then <math>\mathfrak g</math> has the decomposition <math>\mathfrak g = \mathfrak n^- \oplus \mathfrak h \oplus \mathfrak n^+</math> where <math>\mathfrak n^{\pm} = \sum_{\alpha > 0} \mathfrak{g}_{\pm \alpha}</math>. Then <math>\mathfrak b = \mathfrak h \oplus \mathfrak n^+</math> is the Borel subalgebra relative to the above setup. (It is solvable since the derived algebra <math>[\mathfrak b, \mathfrak b]</math> is nilpotent. It is maximal solvable by a theorem of Borel–Morozov on the conjugacy of solvable subalgebras.)

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

See also

• Borel subgroup
• Parabolic Lie algebra

References

• Шаблон:Citation.
• Шаблон:Citation.
• Шаблон:Citation.