Download Commutative Implies Associative?

MASS Colloquium
Commutative Implies Associative?
Notes of the talk by Prof. John Roe
By Bobby Lumpkin and Todd Fenstermacher
Abstract
By introducing the symbol i, with i2 = −1, one can pass from the field of real
numbers to the larger field of complex numbers. In the 19th century various attempts
were made to define still larger “generalized number” fields, such as the quaternions
and octonions, but all of these sacrifice some of the familiar “laws” of arithmetic:
the quaternions are no longer commutative, the octonions not even associative.
Notice that the commutative law apparently “dies” first. Around 1940, Heinz Hopf
made an investigation of generalized number systems that were commutative but
not necessarily associative, and he found that the reals and the complexes are the
only examples. In other words, the commutative law implies the associative law
(in the context in which he was working). Hopf’s methods are topological, and are
closely related to developments in topology in the latter half of the 20th century.
1 Motivation
Professor Roe initiated the talk with some familiar definitions and motivating ideas,
so it seems appropriate to start this paper the same way.
Definition 1.1. (commutative, associative): Given a set A and a binary operation
∗ on the elements of A, we say that ∗ is commutative if for all a, b ∈ A, a ∗ b = b ∗ a.
We say that ∗ is associative if for all a, b, c ∈ A, a ∗ (b ∗ c) = (a ∗ b) ∗ c.
Most of the constructs with a binary operation that we are familiar with possess
both of these nice properties. For example, both R, and C with their usual addition
or multiplication are both commutative and associative.
In the 15th century, mathematicians started generalizing past the real numbers,
producing “new” number systems. We have written new in quotations here since
from the viewpoint of mathematical platonists (in the classical sense), these number
systems exist independently of us. The first of these generalizations was C, the complex number system, which was conceived not so much with the aim of developing a
number system containing a square root of negative one, but as a byproduct of a desire to solve cubic equations. Namely, because in certain cases (“casus irreducibilis”)
complex expressions appear when attempting to solve cubic equations, even though
the roots of the cubic are real. We now recall the definition of C.
Definition 1.2. (Complex Numbers): The set of complex numbers, C, is the set
{a + bi | a, b ∈ R}. Addition is done component (real and imaginary parts) wise and
multiplication accords with multiplication of polynomials in R[i] with i2 = −1.
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Intuitively, we may think of multiplication by a complex number as the composition of a homethety and rotation. Thus, if we consider the natural correspondence
between the complex plane and R2 , multiplication by z ∈ C corresponds to the
linear transformation represented by the matrix
cos(θ) −sin(θ)
ρ
sin(θ) cos(θ)
where ρ is the magnitude of z and θ is arg(z).
In the 19th century, mathematicians became curious with possible iterations of
extending R, to produce “super-complex” numbers from C and so on. In reference
to this progression, Professor Roe recalls the saying, “There is no problem on earth,
however complex, which, if you look at it in the right way, you can not make even
more complex.” In this vein, we get higher dimensional (over R) number systems
such as the quaternions, H, the octonions, O, the sedenions, S, etc. We briefly recall
the structure of the quaternions.
Definition 1.3. (Quaternions): The set of quaternions, H, is the set
{a+bi+cj +dk | a, b, c, d ∈ R}. Addition is done component wise, and multiplication
accords with multiplication of polynomials in R[i, j, k] with i2 = j 2 = k 2 = ijk = −1.
The quaternions, developed by William Rowan Hamilton, may be thought of as a
four dimensional vector space over R. However, unlike in R and C, multiplication is
no longer commutative. This phenomena, of losing nice properties as the dimension
of the extension increases, continues and is one of the themes of the talk.
Dimension
1 R
2 C
4 H
8 O
16 S
Commutative
yes
yes
no
no
no
Associative
yes
yes
yes
no
no
Zero Divisors
no
no
no
no
yes
You may be wondering why certain dimensional cases, such as 3, are notably
absent from the above chart. It turns out that the only number systems, of the sort
we are considering, that have “sensible” properties are those of dimension 2n . From
the above chart, we can also verify that multiplication in these number systems seems
to get progressively worse as the dimension increases. Professor Roe compared this
to the moral decay of civilization, from the blissful innocence of R and C to the sinful
abnormalities of O or S. One observation, in particular, is central to the purpose of
the talk. This is, that commutativity is the first property to die out. In reference
to this, Professor Roe said, “Loosing commutativity is like eating the apple” - again
likening the progression of extensions to the moral decline of civilization.
Of course, this naturally leads one to ask, “Is there a number system that is
commutative, but not associative?”. Well, if we are willing to consider “stupid”
examples, then, yes.
Example 1.1. (Stupid Example): We consider (C, ∗) where ∗ is defined as follows.
a ∗ b = ab. Hence, a ∗ b = b ∗ a, but a ∗ (b ∗ c) = abc 6= abc. This 6= is only in general.
It is perfectly possible that equality may hold in particular instances.
Thus, (C, ∗) is commutative, but not associative. However, this system is “stupid”
because it possesses no unit, since 1 ∗ a = a. It turns out that every commutative
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system (in the sense we have been considering) with a unit is associative, and the
rest of the talk was devoted to explaining this property.
2 Results
We now present the main results from the talk.
Theorem 2.1. (Hopf ): If A is a finite dimensional unital real division algebra, and
is commutative, then it is associative; in fact, either A = R or A = C with their
normal multiplication.
Definition 2.1. (finite dimensional unital real division algebra): We call a number
system, A, a unital real division algebra, if as a vector space over R it is finite
dimensional, A contains a unit element, 1, and A contains an inverse of every one
of its non-zero elements.
Before we are ready to explain the idea behind the proof of this theorem, we
take what may seem like a surprising detour into topology. In particular, we look
at properties of the Möbius Band, as did Professor Roe in his talk. In fact, at this
point in the talk, Professor Roe presented to the audience a model of the Möbius
Band he had constructed out of paper towels, earlier that day. The model even
had a drawing of a face, presumably that of Möbius, on it which had been made in
marker. We now recall a definition of the Möbius Band.
Definition 2.2. (Möbius Band): Topologically, we define the Möbius Band to be
I 2 ∼ where I is the closed unit interval and ∼ is the relation defined by
(x, 0) ∼ (1 − x, 1).
Intuitively, all we have done is to take the unit square, twist one end 180 degrees
and then connect that end to its opposite end. We know that the Möbius Band is
non-orientable. To demonstrate this, professor Roe used his prop. He held up his
model with Möbius’ face right side up, and picked a point on the drawing. He then
followed a continuous path around the center of the band from this starting point,
and when he returned to Möbius, his face was upside down. We also know that
the Möbius Band has only one edge, and hence, that if we were to split the band
down it’s center, we would still wind up with one connected piece. Professor Roe
demonstrated this by making such a cut in his model using a pair of scissors.
Now, we make another important observation about the edge of the Möbius
Band. Namely, that the edge of the Möbius Band is like a circle which winds
around twice, so we will refer to it as S 1 . With this in mind, we see that the Möbius
Band instantiates a mapping from S 1 to itself. In particular, the map to which we
are referring is the map that takes a point on the edge and projects it onto the line
running through the center of the band, which we may also regard as S 1 . For our
purposes, and that of the talk, this is an important map, and we will see shortly
that in fact S 1 is the only sphere which has such a map. We call this map 2 : 1,
since it sends two points on the edge to the same point on the center line. However,
this is not just any old 2 : 1 map; it is a map which respects continuity, a crucial
fact since in topology we are interested in continuous actions. To demonstrate a
discontinuous action, Professor Roe excitedly ripped his model Möbius Band into
pieces and threw them on the floor. Moreover, this map is a 2 : 1 covering map.
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Definition 2.3. (covering map): Let p : Y 7→ X be continuous and surjective
(X, Y topological spaces). If every point x ∈ X has a neighborhood U such that
p−1 (U ) is the union of disjoint open sets Vα ⊂ Y such that P restricted to Vα is a
homeomorphism of Vα onto U for each α, then we call p a covering map, and Y is
said to be a covering space of X.
We say that a covering map, ρ : X 7→ Y is two to one, denoted 2 : 1, if for each
y ∈ Y , |ρ−1 (y)| = 2, i.e. its pre-image is two distinct points in X. This pre-image
is referred to as the fiber of y. We briefly recall a definition of the n-dimensional
sphere, and present the key result involved in proving Hopf’s theorem.
Definition 2.4. (S n ): The n-dimensional sphere, S n , is the set
{(x0 , ..., xn ) ∈ Rn+1 | x20 + x21 + ... + x2n = 1}.
Theorem 2.2. S 1 is the only sphere that has a 2 : 1 connected covering.
Note: Given any space X it is trivial to construct a disconnected 2 : 1 covering.
Simply take the space, Y , consisting of two disjoint copies of X.
Proof. (sketch) Let C be a 2 : 1 connected covering space of S n with n > 1 under the
map p. Consider a point x ∈ S n and the two points c0 , c1 in its pre-image. Since S n
is an n-dimensional manifold, and covering spaces are locally homeomorphic to their
base space, C is an n-dimensional manifold, and thus, connectedness implies path
connectedness. Consider a path, γ, from c0 to c1 . Then p(γ) is a closed loop in S n
based at x. All closed loops in S n can be “shrunk” continuously to the identity path
(i.e. S n has trivial fundamental group) for n > 1. Thus, if we lift this continuous
shrinking, or homotopy, into C, we get a continuous change from the path γ with
two distinct endpoints, to the identity path on c0 (one endpoint). But, of course,
no such continuous change exists (since you would have to “break” from c1 at some
point). This is a contradiction.
This is very nice topological theorem, but how does it tie in to our investigation
about commutativity and associativity in unital real division algebras? Let A be
some commutative finite dimensional unital real division algebra, and A∗ = A \ {0}.
Consider q : A∗ 7→ A∗ given by q(x) = x2 . Since q may not be onto, we restrict
our attention to its image and note that it is a 2 : 1 covering map from A∗ to
its image. Why this is a 2 : 1 map is not entirely obvious, and is where we use
our assumption of commutativity. For example, this is not the case in H since
i2 = j 2 = k 2 = −1. Suppose a is in the image of q, so that its pre-image is nonempty. Then the pre-image of a precisely consists of the roots of x2 − b2 , where b
is an element of the pre-image. However, since our operation is commutative, this
polynomial may always be factored as (x − b)(x + b), and thus can have at most two
roots. Now we consider two possibilities; either A is 1-dimensional over R, or it is of
higher dimension. In the first case, A = R, and in the second case, q is a connected
covering map and it is onto. In topology, often times punctured n-space and S n are
considered to be equivalent (homotopy equivalent), so q is really a 2 : 1 connected
covering map from S n−1 to S n−1 , which, by our previous theorem, implies n = 2,
in which case, A = C which is associative, proving Hopf’s theorem. This was the
conclusion of the talk.
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