Borel–Bott–Weil theorem
Encyclopedia
In mathematics
, the Borel–Weil-Bott theorem is a basic result in the representation theory
of Lie group
s, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundle
s, and, more generally, from higher sheaf cohomology
groups associated to such bundles. It is built on the earlier Borel–Weil theorem
of Armand Borel
and André Weil
, dealing just with the section case, the extension being provided by Raoul Bott
. One can equivalently, through Serre's GAGA
, view this as a result in complex algebraic geometry in the Zariski topology
.
Lie group or algebraic group
over
, and fix a maximal torus
T along with a Borel subgroup
B which contains T. Let λ be an integral weight
of T; λ defines in a natural way a one-dimensional representation Cλ of B, by pulling back the representation on T = B/U, where U is the unipotent radical of B. Since we can think of the projection map G → G/B as a principal B-bundle
, for each Cλ we get an associated fiber bundle L-λ on G/B (note the sign), which is obviously a line bundle
. Identifying Lλ with its sheaf
of holomorphic sections, we consider the sheaf cohomology
groups
. Since G acts on the total space of the bundle 
by bundle automorphisms, this action naturally gives a G-module structure on these groups; and the Borel–Weil–Bott theorem gives an explicit description of these groups as G-modules.
We first need to describe the Weyl group
action centered at
. For any integral weight 
and 
in the Weyl group W, we set 
, where 
denotes the half-sum of positive roots of G. It is straightforward to check that this defines a group action, although this action is not linear, unlike the usual Weyl group action. Also, a weight 
is said to be dominant if 
for all simple roots 
. Let 
denote the length function on W.
Given an integral weight
, one of two cases occur: (1) There is no 
such that 
is dominant, equivalently, there exists a nonidentity 
such that 
; or (2) There is a unique 
such that 
is dominant. The theorem states that in the first case, we have
for all i;
and in the second case, we have
for all 
, while
is the dual of the irreducible highest-weight representation of G with highest weight 
.
It is worth noting that case (1) above occurs if and only if
for some positive root 
. Also, we obtain the classical Borel–Weil theorem
as a special case of this theorem by taking
to be dominant and 
to be the identity element 
.
, an integral weight is specified simply by an integer n, and ρ = 1. The line bundle Ln is O(n), whose sections are the homogeneous polynomials of degree n (i.e. the binary form
s). As a representation of G, the sections can be written as Symn(C2)*, and is canonically isomorphic to Symn(C2). This gives us at a stroke the representation theory of
: Γ(O(1)) is the standard representation, and Γ(O(n)) is its n-th symmetric power. We even have a unified description of the action of the Lie algebra, derived from its realization as vector fields on the Riemann sphere: if H, X, Y are the standard generators of 
, then we can write
of characteristic
. Then it remains true that 
for all i if 
is a weight such that 
is non-dominant for all 
. However, the other statements of the theorem do not remain valid in this setting.
More explicitly, let
be a dominant integral weight; then it is still true that 
for all 
, but it is no longer true that this G-module is simple in general, although it does contain the unique highest weight module of highest weight 
as a G-submodule. If 
is an arbitrary integral weight, it is in fact a large unsolved problem in representation theory to describe the cohomology modules 
in general. Unlike over 
, it need not be the case for a fixed 
that these modules are all zero except in a single degree i.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Borel–Weil-Bott theorem is a basic result in the representation theory
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 studiesmodules over these abstract algebraic structures...
of Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s, and, more generally, from higher sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
groups associated to such bundles. It is built on the earlier Borel–Weil theorem
Borel–Weil theorem
In mathematics, in the field of representation theory, the Borel–Weil theorem, named after Armand Borel and André Weil, provides a concrete model for irreducible representations of compact Lie groups and complex semisimple Lie groups. These representations are realized in the spaces of global...
of 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...
and André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
, dealing just with the section case, the extension being provided by Raoul Bott
Raoul Bott
Raoul Bott, FRS was a Hungarian mathematician known for numerous basic contributions to geometry in its broad sense...
. One can equivalently, through Serre's GAGA
Gaga
- Entertainment :* Lady Gaga , American recording artist* Gaga , a dancing technique* Rara, a type of Caribbean music called Gaga in the Dominican Republic* GNZ-004 Gaga, a mobile weapon in Mobile Suit Gundam 00- Sports :...
, view this as a result in complex algebraic geometry in the Zariski topology
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...
.
Formulation
Let G be a semisimpleSemisimple algebraic group
In mathematics, especially in the areas of abstract algebra and algebraic geometry studying linear algebraic groups, a semisimple algebraic group is a type of matrix group which behaves much like a semisimple Lie algebra or semisimple ring.- Definition :...
Lie group or algebraic group
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...
over
, and fix a maximal torusMaximal 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...
T along with a Borel subgroup
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 group GLn ,...
B which contains T. Let λ be an integral weight
Weight (representation theory)
In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F – a linear functional – or equivalently, a one dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group...
of T; λ defines in a natural way a one-dimensional representation Cλ of B, by pulling back the representation on T = B/U, where U is the unipotent radical of B. Since we can think of the projection map G → G/B as a principal B-bundle
Principal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
, for each Cλ we get an associated fiber bundle L-λ on G/B (note the sign), which is obviously a line bundle
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
. Identifying Lλ with its sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of holomorphic sections, we consider the sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
groups
. Since G acts on the total space of the bundle by bundle automorphisms, this action naturally gives a G-module structure on these groups; and the Borel–Weil–Bott theorem gives an explicit description of these groups as G-modules.We first need to describe the Weyl group
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection...
action centered at
. For any integral weight and in the Weyl group W, we set , where denotes the half-sum of positive roots of G. It is straightforward to check that this defines a group action, although this action is not linear, unlike the usual Weyl group action. Also, a weight is said to be dominant if for all simple roots . Let denote the length function on W.Given an integral weight
, one of two cases occur: (1) There is no such that is dominant, equivalently, there exists a nonidentity such that ; or (2) There is a unique such that is dominant. The theorem states that in the first case, we have for all i;and in the second case, we have
for all , while is the dual of the irreducible highest-weight representation of G with highest weight .It is worth noting that case (1) above occurs if and only if
for some positive root . Also, we obtain the classical Borel–Weil theoremBorel–Weil theorem
In mathematics, in the field of representation theory, the Borel–Weil theorem, named after Armand Borel and André Weil, provides a concrete model for irreducible representations of compact Lie groups and complex semisimple Lie groups. These representations are realized in the spaces of global...
as a special case of this theorem by taking
to be dominant and to be the identity element .Example
For example, consider G = SL2(C), for which G/B is the Riemann sphereRiemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
, an integral weight is specified simply by an integer n, and ρ = 1. The line bundle Ln is O(n), whose sections are the homogeneous polynomials of degree n (i.e. the binary form
Binary form
Binary form is a musical form in two related sections, both of which are usually repeated. Binary is also a structure used to choreograph dance....
s). As a representation of G, the sections can be written as Symn(C2)*, and is canonically isomorphic to Symn(C2). This gives us at a stroke the representation theory of
: Γ(O(1)) is the standard representation, and Γ(O(n)) is its n-th symmetric power. We even have a unified description of the action of the Lie algebra, derived from its realization as vector fields on the Riemann sphere: if H, X, Y are the standard generators of , then we can writePositive characteristic
One also has a weaker form of this theorem in positive characteristic. Namely, let G be a semisimple algebraic group over an algebraically closed fieldAlgebraically 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:...
of characteristic
. Then it remains true that for all i if is a weight such that is non-dominant for all . However, the other statements of the theorem do not remain valid in this setting.More explicitly, let
be a dominant integral weight; then it is still true that for all , but it is no longer true that this G-module is simple in general, although it does contain the unique highest weight module of highest weight as a G-submodule. If is an arbitrary integral weight, it is in fact a large unsolved problem in representation theory to describe the cohomology modules in general. Unlike over , it need not be the case for a fixed that these modules are all zero except in a single degree i.