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In mathematics, specifically in
representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
, a Borel subalgebra of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
\mathfrak is a maximal solvable subalgebra. The notion is named after
Armand Borel Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in alg ...
. If the Lie algebra \mathfrak is the Lie algebra of a
complex Lie group In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G \times G \to G, (x, y) \mapsto x y^ is holomorphic. Basic examples are \operatorname_n(\math ...
, then a Borel subalgebra is the Lie algebra of a
Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgr ...
.


Borel subalgebra associated to a flag

Let \mathfrak g = \mathfrak(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 A flag is a piece of textile, fabric (most often rectangular) with distinctive colours and design. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design employed, and fla ...
of ''V''; given a flag V = V_0 \supset V_1 \supset \cdots \supset V_n = 0, the subspace \mathfrak b = \ is a Borel subalgebra, and conversely, each Borel subalgebra is of that form by
Lie's theorem In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if \pi: \mathfrak \to \mathfrak(V) is a finite-dimensional representation of a solvable Lie algebra, then ...
. 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 In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of Simple Lie algebra, simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals ...
, \mathfrak h a
Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
and ''R'' the
root system In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and ...
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^ = \sum_ \mathfrak_. 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/math> 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^+. It is the same thing as a \mathfrak b-weight vector (Proof: if h \in \mathfrak h and e \in \mathfrak^+ with
, e The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
= 2e and if \mathfrak \cdot v is a line, then 0 =
, e The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
\cdot v = 2 e \cdot v.)


See also

*
Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgr ...
*
Parabolic Lie algebra In algebra, a parabolic Lie algebra \mathfrak p is a subalgebra of a semisimple Lie algebra \mathfrak g satisfying one of the following two conditions: * \mathfrak p contains a maximal solvable subalgebra (a Borel subalgebra) of \mathfrak g; * th ...


References

*. *. *. Representation theory Lie algebras {{algebra-stub