Borel subgroup
Encyclopedia
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 group GLn (n x n invertible matrices),
the subgroup of invertible upper triangular matrices is a Borel subgroup.
For groups realized over algebraically closed field
s,
there is a single conjugacy class
of Borel 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.
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
es of parabolic subgroups is in bijection with the set of all subsets of nodes of the corresponding Dynkin diagram; 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.)
Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive
) algebraic groups, in Jacques Tits
' 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
contained in B.
The notion was introduced by Armand Borel
, who played a leading role in the development of the theory of algebraic groups.
with a Cartan subalgebra 
, given an ordering
of
, the Borel subalgebra 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
.
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
G is a maximal Zariski closed and connected
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
solvable
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
algebraic subgroup.
For example, in the group GLn (n x n invertible matrices),
the subgroup of invertible upper triangular matrices is a Borel subgroup.
For groups realized over algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
s,
there is a single conjugacy class
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
of Borel 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.
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
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
es of parabolic subgroups is in bijection with the set of all subsets of nodes of the corresponding Dynkin diagram; 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.)
Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive
Reductive group
In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...
) algebraic groups, in Jacques Tits
Jacques Tits
Jacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group.- Career :Tits received his doctorate in mathematics at the age of 20...
' 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
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 Lie group G is a compact, connected, abelian Lie subgroup of G . A maximal torus is one which is maximal among such subgroups...
contained in B.
The notion was introduced by Armand Borel
Armand Borel
Armand Borel 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...
, who played a leading role in the development of the theory of algebraic groups.
Lie algebra
For the special case of a Lie algebraLie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
with a Cartan subalgebra , given an orderingOrder theory
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
of
, the Borel subalgebra 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 algebraParabolic 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 of \mathfrak g;...
.