Borel Subalgebra
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In mathematics, specifically in
representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, a Borel subalgebra of a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
\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(\mat ...
, 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 subgroup ...
.


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 fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design empl ...
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, the ...
. Hence, the Borel subalgebras are classified by the
flag variety In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smo ...
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 simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra i ...
, \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 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 representati ...
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= 2e and if \mathfrak \cdot v is a line, then 0 = , e\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 subgroup ...
*
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; * the ...


References

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