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# A brief survey of symmetry in mathematics

This paper presents a brief survey of the idea of symmetry in mathematics, as exemplified by some particular developments in algebra, differential equations, topology, and number theory.

At this conference on Symmetries Throughout the Sciences, I thought that it might be of interest to survey briefly how the idea of symmetry has developed in its native habitat—mathematics.

## Section 1

Every congruence of two triangles in the Euclidean plane gives a
symmetry of the plane. But millenia passed before mathematicians began
to consider the totality of symmetries *G* of the plane. Once
that is done, one sees that there is an algebraic operation on
*G* combining two elements to get a third, namely the
composition *g*·*h*, which sends any point *x* to
*g*(*h*(*x*)). If one extracts the notion of group from this
context, a group *G* is a set with a binary operation
(*g*_{1}, *g*_{2}) →
*g*_{1}·*g*_{2}, which is associative:
(*g*_{1}·*g*_{2})·*g*_{3} =
*g*_{1}·(*g*_{2}·*g*_{3}), has an
identity element 1 such that *g*·1 = 1·*g* for
all *g* in *G*, and each *g* in *G*
has an inverse *g*^{−1} such that
*g*^{−1}·*g* = 1 =
*g*·*g*^{−1}. Towards the end of the 18^{th}
century, this idea was used by J. L. Lagrange, A. T. Vandermonde, and
P. Ruffini. One of the great advances in mathematics was the
exploitation of this new notion by E. Galois (1811–1832) in an
algebraic context.

Let *f*(*x*) = *x*^{n} + *a*_{1}*x*^{n−1}
+ … + *a*_{n} be a polynomial with coefficients in
the *field* ℚ of rational numbers {*m*/*n*; *m*, *n*
∈ ℤ, the ring of integers}. Then with the use of complex
numbers, *f*(*x*) = ∏_{i=1}^{n} (*x* −
θ_{i}) factors into linear factors,
θ_{i} being the roots of *f*(*x*).

Galois considered the group of all permutations of the roots
θ_{1}, … , θ_{n}, which respect
addition and multiplication of all expressions formed by them together
with rational numbers in the *field* they generate.

This simple idea has had far reaching applications. To give a
rudimentary example, consider the field generated by the roots of
Let ζ = cos 2π/8 + *i* sin 2π/8
= /2 + *i* /2. Then the zeros of
*x*^{8} − 1 are 1, ζ, … , ζ^{7},
and ζ satisfies *x*^{4} + 1 = 0. The group
*G* of automorphisms of the field ℚ(ζ) consists of the
four automorphisms
where *gcd*(*n*, 8) = 1, i.e., *n* = 1, 3, 5, 7.

Each element of the Galois group *G* other than
σ_{1} has order 2; since
Then {σ_{1}, σ_{n}} forms a
subgroup of two elements ≈ ℤ/2ℤ, *n* = 3, 5, 7
*G* has three subgroups different from σ_{1}.

In the field ℚ(ζ)(= ℚ[*i*, ]),
That is, to *each subgroup H* of *G* there
*corresponds the unique subfield* of ℚ(ζ) whose elements
it fixes, and conversely each subfield *F* of ℚ(ζ) arises
in this way. Finally, the number of elements in *G* =
dim(ℚ(ζ), the dimension of ℚ(ζ) as a vector space with
scalars ℚ.

These relations between a field generated by all the roots of a
polynomial in ℚ[*x*] and its Galois group hold generally.

Galois introduced the notion of normal subgroup (he called it “propre”), proved that an irreducible polynomial of degree 5 or greater cannot be expressed by radicals, and exhibited such irreducible polynomials. N. Abel (1802–1829) proved a similar result but only for “general polynomials” with indeterminate coefficients.

As an even easier application, one could settle the long-standing geometric conjectures about the impossibility of angle trisection and cube doubling via compass and ruler construction.

The interest in crystallography led to the study of a class of infinite
groups. M. L. Frankenheimer in 1842 classified all the
crystallographic lattices in ℝ^{3}; these are groups Γ,
which contain a subgroup *T* of translations in three
independent directions in 3-space, and moreover the set of cosets
{*xT*; *x* ∈ Γ}, which is denoted
Γ/*T*, is a finite set. Indeed, if we take *T* to
be the subgroup of *all* translations in Γ, then
*xTx*^{−1} = *T* and thus Γ/*T* is a
finite group. In 1850, Auguste Bravais reworked Frankenheimer’s
results more rigorously, discovering that there were only 14 types of
crystallographic groups in ℝ^{3}, correcting
Frankenheimer’s claim of 15. These are known as *Bravais
lattices.*

We’ll come back to lattices later.

## Section 2

Let us turn now to “finite continuous” groups—these contain
an infinity of elements parametrized by a finite number of parameters.
The first major achievement was by the Norwegian mathematician, Sophus
M. Lie (1842–1899), who in 1873–1874 associated to each such group
its algebra of infinitesimal generators. By definition, an
*infinitesimal generator *in local coordinates
(*x*_{1}, … , *x*_{n}) is an operator
which describes how the smooth function *f* changes at
the points *x* in the direction
(ξ^{1}(*x*), … , ξ^{n}(*x*)).

Geometrically, *X* defines a *vector field*, which we
can think of as the velocity field of the flow
*dx*^{i}/*dt* = ξ^{i}(*x*)(*i* =
1, … , *n*). The resulting flow after time *t*
results in the “finite transformation” denoted *exp **t*
*X*, which moves each point to its downstream position after time
*t*. This general notion agrees with the usual exponential:

If *A* is any *n* × *n* matrix and ξ(*x*) =
*Ax* for *x* ∈ ℝ^{n}, then
*exp* *t* *A* = *e*^{tA}.

Lie’s earliest interest was in getting quadratures for differential
equations from symmetries. Later, he studied the group structures. If
*G* is a “finite continuous” group, with *u*, *v*,
*w* in *G* and *w* = *u*·*v*, then the group
multiplication law is specified by *r* functions of
2*r* variables, *r* = dim*G*. The big
idea of Lie was that group multiplication can be determined by merely
*r*^{3} numbers in the following way.

A finite continuous transformation group *G* on a space
*M* is specified by:

(*i*) smooth manifolds *G* and *M*,

(*ii*) a *multiplication **G* × *G* →
*G*:(*g*_{1}, *g*_{2}) → *g*_{1}*g*_{2},

(*iii*) an *action **G* × *M* → *M*:(*g*, *m*) →
*g*(*m*) of *G* on *M* such that
(*g*_{1}*g*_{2})(*m*) =
*g*_{1}(*g*_{2}(*m*)).

Let *c*(*t*) be any smooth curve in *G* with *c*(0)
= 1. Then *m* → *dc*(*t*)(*m*)/*dt*_{|t=0} is a
vector field on *M*. Let 𝔊 denote the totality of such
vector fields. In the special case where *G* acts on itself by
left multiplication, we call 𝔊 the Lie algebra of *G*; it is
often denoted Lie(*G*).

## Properties:

1. *X*, *Y* ∈ 𝔊 ⇒ *X* + *Y* ∈ 𝔊,
*aX* ∈ 𝔊, for all *a* ∈ ℝ.

2. *X*, *Y* ∈ 𝔊 ⇒ [*X*, *Y*] ∈ 𝔊
(where for any function *f* on *M*, [*X*,
*Y*]*f* = *XYf* − *YXf*).

3. [*X*[*Y*, *Z*]] + [*Z*, [*X*, *Y*]] +
[*Y*[*Z*, *X*]] = 0.

4. dim 𝔊 ≤ dim *G*, with equality if *g*(*m*) =
*m* for all *m* ∈ *M* implies *g* = 1.

Let {*X*_{1}, … , *X*_{r}} be a
base for *Lie(G)*. Then [*X*_{i},
*X*_{j}] = Σ_{k=1}^{r} *c*_{i,j}^{k} *X*_{k}, *c*_{i,j}^{k} ∈ ℝ.

The *r*^{3} constants
*c*_{i,j}^{k} are called Lie’s constants of
structure.

Lie introduced the notion of a *simple* group
*G*—one having no normal subgroup of positive dimension
≠ *G*. He listed four “great” classes of simple groups
of *rank n:
*
These are the matrix groups that preserve det,
*x*_{1}^{2} + … + *x*_{2n+1}^{2}, a
skew-symmetric bilinear form in *x*_{1}, … ,
*x*_{2n}, and *x*_{1}^{2} + … +
*x*_{2n}^{2}, respectively. He had in mind the group with
complex entries. The rank *n* refers to the dimension of any
maximal diagonalizable subgroup of *G*—all such being
conjugate under an inner automorphism (Fig. 1).

W. Killing in 1888–1889, classified all simple Lie algebras (over ℂ)
of rank *n* (1, 2). His results needed a slight correction,
which Cartan did in this 1896 thesis. Hermann Weyl put Cartan’s
classification into a more geometric form, which permits us to give a
remarkably vivid picture of the classification.

To each simple Lie group, there corresponds a unique *simply
connected *simple Lie group *G* (i.e., any circle in
*G* can be deformed to a point.) Each simple Lie group
*G* over ℂ has a maximal compact subgroup
*G*_{K} unique up to conjugacy. Lie
(*G*_{K}) is a real Lie algebra and
*Lie*(*G*) = *Lie*(*G*_{K}) ⊗
ℂ; that is, the complex Lie algebra *Lie*(*G*)
consists of complex Lie combinations of elements of
*Lie*(*G*_{K}).

Choose *T*, a maximal diagonalizable subgroup of
*G*_{K} (all such *T* are conjugate in
*G*_{K}). *G* acts on itself by inner automorphisms.
This action induces a linear action on the Lie algebra 𝔊 called the
adjoint representation.

Decompose 𝔊 = ⊕𝔊_{α} into spaces of common
eigenvectors for *Ad* *T*, i.e., *Ad*(*t*)*X*_{α} =
α(*t*)*X*_{α} for all *X*_{α} ∈
𝔊_{α}, *t* ∈ *T*.

Let *L* = Ker exp: Lie(*T*) → *T*; i.e.,
*x* ∈ *L* ↔ *x* ∈ *Lie*(*T*) and
*expX* = 1. Then *L* is a lattice in the sense
of crystallography.

Let *N*(*T*) = {*g* ∈ *G*; *gTg*^{−1} = *T*}. Then
*N*(*T*) acts on *Lie(T)* and stabilizes *L*.
Moreover, *W* = *N*(*T*)/*T* acts faithfully on *L*.

Γ = *W* × *L* (semi-direct product) is a
crystallographic group; we call it “highly symmetric”, by which
we mean that Γ is generated by reflections in the *n* +
1 faces of an *n*-dimensional simplex (Fig.
1).

The classification of simple Lie groups can be formulated as: *The
correspondence
*
*is a* 1 − 1 *correspondence*. Indeed all
the information about the algebraic and geometric structure of
*G* can be deduced from the crystal structure (1).

The early interest of S. Lie in applications of infinitesimal symmetries to differential equations has borne fruit. Well known to physicists is the theorem of E. Noether (1918) that to each infinitesimal symmetry there corresponds a conservation law (3).

Starting with a paper of Date *et al*. (4), a remarkable link
was discovered between representation theory of affine Lie algebras and
the Korteweg–deVries nonlinear partial differential equations.
Starting in 1978, it has been realized that complete integrability is
related to Lie algebra theory and that both Korteweg–deVries equations
and Toda systems can be viewed as Hamiltonian systems in the co-adjoint
orbit of a suitable Lie algebra (2, 5, 6).

## Section 3

The most important application of groups to topology lies in the
construction via groups of special space prototypes in terms of which
all spaces can be analyzed. The construction is via *quotient
spaces whose points are cosets of a group.*

We consider in this section quotients of compact groups. In the next section, we consider quotients of noncompact groups by discrete subgroups.

A fiber bundle (*E*, *B*, *F*, π, *G*) with total space
*E*, base *B*, fiber *F*, projection
π:*E* → *B*, and group *G* is specified by a
collection of open sets {*V*_{i}} covering
*B* and by homeomorphisms for each index *i*:
satisfying
where *g*_{ji}(*v*) ∈ *G* and is continuous in
*v* ∈ *V*_{i} ∩ *V*_{j}.

For example, a möbius strip is the total space of a fiber bundle
over the circle *S*^{1}, with fiber the unit interval
*I* = [0, 1] and the group *G* =
ℤ/(2) acting by flipping *I*.

If in Eq. **1** we replace the fiber *F* by the group
*G* acting on itself by left multiplication, we obtain the
“principal” *G*-bundle associated to (*E*, *B*, *F*,
π, *G*), which we denote (*P*, *B*, *G*, π). Since left and
right multiplication in any group commute, the action of *G*
on *P* by right multiplication can be defined unambiguously. A
principal *G*-bundle (*P*_{G}, *B*_{G}, *G*,
π_{G}) is called *n*-universal, if and only if
π_{k}(*P*_{G}) = 0, 0 ≤ *k* < *n*;
i.e., any map of the *k*-sphere *S*^{k} to
*P*_{G} can be deformed to a point for all
*k* < *n*. Its basic property is: For any principal
*G*-bundle (*P*, *B*, *G*, π) with *dimB* ≤
*n*, there is a *G*-bundle map f:(*P*, *B*, *G*, π) →
(*P*_{G}, *B*_{G}, *G*, π_{G}) (i.e., f is
continuous and respects fibers and hence defines a map f:*B* →
*B*_{G}) and therefore the bundle (*P*, *B*, *G*, π)
is equivalent to the *pull-back bundle*, defined as
We can take as *N*-universal *U*(*q*) bundle the
principal bundle (*P*_{G}, *B*_{G}, *U*(*q*),
π_{G}) with
Clearly dim_{ℂ} *Gr*(*q*, *N*, ℂ) = *qN*.

Fix ℂ^{i+N−i} in ℂ^{q+N}.
Let
𝒵_{i} is called a Schubert cycle;
dim_{ℝ} 𝒵_{i} = 2(*qN* − *i*).
Intersections of 𝒵_{i} with cycles of dim
2_{i} defines a linear function on the homology
group *H*_{2i}(*Gr*(*q*, *N*, ℂ), ℤ); hence, an element
*c*_{i} the cohomology group
*H*^{2i}(*Gr*(*q*, *N*, ℤ)).

The *i*^{th} Chern class of principal bundle
(*P*, *B*, *G*, π) is defined as the element in
*H*^{2i}(*B*, ℤ) given by
where *N* is taken ≥ dim *B*.

These topological invariants can be expressed in terms of the curvature of the bundle

*Example*. (*P*, *B*, *G*, π) with
*G* = *GL*(*q*, ℂ)

φ = connection on bundle = 1-form, values in *Lie*(*G*),

Φ = curvature = *d*φ − 1/2 [φ, φ] is a
*Lie*(*G*) − valued 2 − form on *p*

det(λ*I*_{q} − 1/(2π) *X*) =
λ^{q} + *F*_{1}(*X*)λ^{q−1} + … +
*F*_{q}(*X*), *x* ∈ *Lie*(*G*).

Then *F*_{i}(Φ) = *c*_{i}(*P*, *B*, *G*, π) ∈
*H*^{2i}(*B*, ℤ) (7).

The foregoing ideas are exploited in Yang–Mills theories.

## Section 4. Discrete Groups

By a lattice Γ in a Lie group *G*, we mean a subgroup
which is

(*i*) discrete (i.e., no accumulation points),

(*ii*) *G*/Γ has finite Haar measure.

*Example. **SL*(*n*, ℤ) is a lattice in
*SL*(*n*, ℝ).

Borel–Harish–Chandra Theorem (1961). G(ℤ)
*is a lattice in *G(ℝ) *for any algebraic group defined
over *ℚ *with no homomorphisms to scalars defined over
*ℚ (8).

Such lattices and closely related ones are called arithmetic.

A near converse to the foregoing theorem is the

Margulis Theorem (1974). *A lattice in a
semi-simple group of ℝ-rank *> 1 and with no ℝ-rank 1
factors is arithmetic (9).

Margulis’ result came from a strengthening of the following rigidity theorem.

Theorem (Mostow, 1972). *Let
*G, G′ *be semi-simple Lie groups having no compact factors
and no centers. Let Γ, Γ′ be lattices in *G, G′,
*respectively. Assume *G/Γ *and *G′/Γ′
*are compact, and θ:Γ → Γ′ is an isomorphism. Then θ
extends to an analytic isomorphism *G → G′, *except in the
case that *G = SL(2, ℝ)/± 1.

Underlying the proof of this rigidity theorem is the study of
the space of double cosets *K*/*G*/Γ associated to the pair
(*G*, Γ), where *K* is a maximal compact subgroup
(10). The space *K*/*G* is a *symmetric Riemannian*
space; at each of its points, the map reversing the direction of
geodesics gives an isometry of the space [Example: hyperbolic
*n*-space = *SO*(*n*)/*SO*_{o}(*n*, 1)].
The space *K*/*G*/Γ for (*G*, *K*) as above and Γ
a discrete subgroup is a *locally* symmetric space.

This rigidity result is false for *SL*(2, *R*)/± 1, as is
well-known from the theory of one complex variable.

Margulis’ strengthening consisted in replacing the hypothesis on Γ′ by a weaker one, permitting θ to be a homomorphism, and implementing the boundary map strategy used in proving rigidity, but replacing the geometric arguments in Mostow’s proof by measure theoretic arguments based on the “multiplicative ergodic theorem.”

The applications of the rigidity theorem that have been made by W.
Thurston to the study of three-dimensional manifolds depend on the 1968
version that was proved for *SO*(1, *n*) using the theory of
quasi-conformal mappings in *n*-space (11). (The first version
assumes that *G*/Γ is compact; subsequently, the general
strategy was extended by Gopal Prasad to the case of arbitrary lattices
in *SO*(1, *n*), (*n* > 1)) (12).

As a consequence of this rigidity theorem, the metric invariants of
*n*-dimensional manifolds (*n* > 2) having
finite volume hyperbolic structure are topological invariants!

Thurston’s central conjecture is that 3-manifolds can be decomposed
canonically into pieces each having geometric structure
*K*/*G*/Γ, a space of double cosets of a group (13). In
some of the most important cases, the geometric structure is finite
volume hyperbolic. For example, the complement of a figure-eight knot
in *S*^{3} (Figs.
2, 3, 4, 5),
or that obtained by (*p*, *q*) Dehn surgery has finite volume
hyperbolic structure if |*p*| > 4 or |*q*| >
1. The proof is based on remarkable geometry-guided computations
(14).

Another line of investigation in topology has been directed at topological characterization of spaces, which admit negatively curved metrics resembling those on negatively curved locally symmetric spaces (15).

## Section 5. Number Theory

We conclude with a bare bones statement of the ongoing Langlands program (16, 17).

The locally compact adele ring 𝔸 of ℚ is defined as
the restricted direct product of the field ℝ and the
*p*-adic completions ℚ_{p} for all
primes *p*. Then ℚ embeds diagonally in 𝔸 and its image is
discrete. Consider *G* = *GL*(*n*), *G*(𝔸) = *GL*(*n*, 𝔸), and
*G*(ℚ) = *GL*(*n*, ℚ), which is discrete in *G*(𝔸).
We define the regular representation *R* of *G*(𝔸)
on *L*^{2} (*G*(ℚ)/*G*(𝔸)) via *R*(*g*)*f*(*x*) =
*f*(*xg*) for *x*, *g* ∈ *G*. A representation π of
*G*(𝔸) is called *automorphic* if and only if π
is irreducible, unitary, and occurs in the decomposition of
*R* or from an analytic continuation of such.

The decomposition alluded to here owes much to profound work of
Harish-Chandra and, later, Langlands (18, 19). Then π decomposes as a
tensor product
with each π_{p} an irreducible
unitary representation of *G*(ℚ_{p}). For all
*p* ∉ some finite set *S*_{π},
π_{p} contains a vector fixed by
*G*(ℤ_{p}). By the representation theory of
ℚ_{p}, π_{p} = π_{p, u},
i.e., depends on a parameter *u* ∈ ℂ^{n} and
is unitary if *u* ∈ ℝ^{n}.
Moreover,
σ being a permutation.

Set *t*(π_{p}) = conjugacy class of
in *GL*(*n*, ℂ)
Given φ:Gal(ℚ̄, ℚ) → *GL*(*n*, ℂ)
continuous (i.e., Kerφ is the fixer of a finite dimensional Galois
extension field *E* of ℚ), there is a finite set of prime
numbers *S*_{ϕ} such that the ideal
ℤ_{Ep} in the ring ℤ_{E} of
algebraic integers in *E* ≠ *P*^{e} … ,
with *e* > 1 (i.e., *p* does not ramify in
*E*) for all *p* ∉ *S*_{φ}. For such
*p* ∉ *S*_{φ}, the Frobenius
*p*^{th} power automorphism of
ℤ_{E}/*P* over ℤ/(*p*) is induced by
an automorphism denoted *Fr*_{p} ∈ Gal(*E*,
ℚ); *Fr*_{p} is uniquely determined up to conjugacy only.

Langlands’ Conjecture.
*This says that *t(π)* carries fundamental
arithmetic information interrelating the behavior of primes under field
extension.*

*Langlands’ Conjecture* has been proved so far only in the
case that φ maps to *GL*(2, ℂ) with solvable image; this
was accomplished for most cases by Langlands and completed by Tunnell.

It should be observed that the representation π has automorphic forms associated with it. Andrew Wiles in his historic proof of the Shimura–Taniyama conjecture (and hence of Fermat’s last theorem), uses the Langlands–Tunnell result to help him show that a certain representation of the Galois group acting on points of finite order of an elliptic curve is associated with an automorphic form (20).

- Copyright © 1996, The National Academy of Sciences of the USA

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