In the theory of
algebraic groups
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
M ...
, a Borel subgroup of an
algebraic group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
''G'' is a maximal
Zariski closed and connected solvable algebraic subgroup. For example, in the
general linear group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
''GL
n'' (''n x n'' invertible matrices), the subgroup of invertible
upper triangular matrices
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...
is a Borel subgroup.
For groups realized over
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
s, there is a single
conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wor ...
of Borel subgroups.
Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally,
reductive) algebraic groups, in
Jacques Tits
Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric.
Life and ...
' theory of groups with a
(B,N) pair. Here the group ''B'' is a Borel subgroup and ''N'' is the normalizer of a
maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.
A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefore ...
contained in ''B''.
The notion was introduced by
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 ...
, who played a leading role in the development of the theory of algebraic groups.
Parabolic subgroups
Subgroups between a Borel subgroup ''B'' and the ambient group ''G'' are called parabolic subgroups.
Parabolic subgroups ''P'' are also characterized, among algebraic subgroups, by the condition that ''G''/''P'' is a
complete variety
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety , such that for any variety the projection morphism
:X \times Y \to Y
is a closed map (i.e. maps closed sets onto closed sets). This can ...
.
Working over algebraically closed fields, the Borel subgroups turn out to be the minimal parabolic subgroups in this sense. Thus B is a Borel subgroup when the homogeneous space G/B is a complete variety which is "as large as possible".
For a simple algebraic group ''G'', the set of
conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wor ...
es of parabolic subgroups is in bijection with the set of all subsets of nodes of the corresponding
Dynkin diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras ...
; the Borel subgroup corresponds to the empty set and ''G'' itself corresponding to the set of all nodes. (In general each node of the Dynkin diagram determines a simple negative root and thus a one-dimensional 'root group' of ''G''---a subset of the nodes thus yields a parabolic subgroup, generated by ''B'' and the corresponding negative root groups. Moreover, any parabolic subgroup is conjugate to such a parabolic subgroup.)
Example
Let
. A Borel subgroup
of
is the set of upper triangular matrices
and the maximal proper parabolic subgroups of
containing
are
Also, a maximal torus in
is
This is isomorphic to the algebraic torus
.
Lie algebra
For the special case 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 ...
with 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 à ...
, given an
ordering
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
of
, the
Borel subalgebra In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra \mathfrak is a maximal solvable subalgebra. The notion is named after Armand Borel.
If the Lie algebra \mathfrak is the Lie algebra of a complex Lie group, ...
is the direct sum of
and the
weight spaces of
with positive weight. A Lie subalgebra of
containing a Borel subalgebra is called a
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 ...
.
See also
*
Hyperbolic group
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a ''word hyperbolic group'' or ''Gromov hyperbolic group'', is a finitely generated group equipped with a word metric satisfying certain properties abstra ...
*
Cartan subgroup
In algebraic geometry, a Cartan subgroup of a connected linear algebraic group over an algebraically closed field is the centralizer of a maximal torus (which turns out to be connected). Cartan subgroups are nilpotent and are all conjugate.
Examp ...
*
Mirabolic subgroup In mathematics, a mirabolic subgroup of the general linear group GL''n''(''k'') is a subgroup consisting of automorphisms fixing a given non-zero vector in ''k'n''. Mirabolic subgroups were introduced by . The image of a mirabolic subgroup in th ...
References
*
*
*
;Specific
External links
*
*{{SpringerEOM, title=Borel subgroup , id=Borel_subgroup , oldid=14476 , first=V.P. , last=Platonov
Algebraic groups