Dean, Frank, Owen,
That would be 3 hours delightfully spent. Sign me up.
Thanks! -
Grant
On 1/24/12 8:21 PM, Frank Wimberly wrote:
This is a message from Dean Gerber. For some reason it didn't reach
the List when he sent it. I forward it at his request. I will
certainly attend the lecture he offers.
Algebras
Owen--
I think what you are looking for is the theory of algebras,
generally known as non-associative algebras. These structures are
vector spaces V(F) defined over a field of scalars F satisfying the
usual axioms of a vector space with respect to the operations of
vector addition and scalar multiplication, and an additional binary
operation of vector multiplication (called a product) that is
distributive with respect the vector space operations. To be specific,
let x,y,z be vectors in V(F), let a,b be scalars in F, and denote the
product of x and y by x!y. Then V(F) and the product (!) define
an algebra if and only if
i) x!y is in V. (x!y is a vector - the very meaning of "binary
composition")
ii) a(x!y) = (ax)!y = x!(ay). ( Scalar multiplication distributes with
vector multiplication)
iii) x!(y + z) = x!y + x!z and (y + z)!x = y!x + z!x. (Vector
multiplication distributes with vector addition)
Since a vector space is always equivalent to a set of tuples, this
provides the multiplication of tuples you are looking for. For an
n-dimensional vector space the generic (general) product is completely
defined by n-cubed parameters, known as the structure constants.
Specific choices of these parameters from the field F define specific
algebras and the properties of these algebras vary greatly over the
possible choices. For example, for n = 2 there are 8 free parameters
and the complex numbers represent a single point in this 8 dimensional
space of structure constants. That particular choice implies that the
algebra of complex numbers is itself field. Generally, algebras have
no properties other than i) to iii) above, i.e. they are generally not
commutative or associative.
The Caley-Dickson procedure is a process by which a "normed" algebra
can be extended to a normed algebra of twice the dimension. The only
real one dimensional normed algebra is the real number field itself.
The Caley-Dickson extension is just the complex numbers as a 2
dimensional algebra, and it is also a field; in fact the only 2
dimensional field..
The Caley-Dickson extension of the complex number algebra is the 4
dimensional quaternion algebra. But, the quaternions are NOT a field:
they are not commutative even though they are associative and a
division algebra. They are often known as a "skew" field,
The Caley-Dickson extension of the quaternions is the 8 dimensional
octonion algebra, and these are neither commutative or associative,
but they are a division algebra.
The next step gives the nonions of dimension 16 at which point we
lose the last semblance of a field because they are not commutative,
not associative, and not a division algebra. Thus, if we want fields,
the complex numbers are indeed the end point. All real division
algebras are of dimensions 1,2,4, or 8! There are many division
algebras in the dimensions 2,4,8, but only in n = 2 are all of them
classified up to isomorphism.
I could go on, if you could gather up an audience of at least ten for
a (free) three hour blackboard lecture with two breaks. For an
audience of fewer than ten I would have to collect ten hours of Santa
Fe minimum wages for prep and lecture time. Its a beautiful subject
with a very colorful history, and includes the quaternions, octonians,
Lie algebras, Jordan algebras, associative algebras, everything
mentioned by the FRIAM commentariat.
Regards- Dean Gerber
*From:*[email protected] [mailto:[email protected]]
*On Behalf Of *Owen Densmore
*Sent:* Tuesday, January 24, 2012 9:34 AM
*To:* Complexity Coffee Group
*Subject:* Re: [FRIAM] Complex Numbers .. the end of the line?
Arlo:
...Would it not be better to say, "are there number(data?)-structures
that provide for interesting algebras not yet considered?"
Yes indeed. I was fumbling for a way to say that but ran out of steam!
Roger Critchlow:
http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
Now that is interesting, and nice to know this is a broader
conversation than I had known. GA's .. gotta look into them and their
unification of complex numbers and vectors.
Roger/Carl:
Suspect you are about to pop out of algebra and end up someplace
else as interesting.
As you say, I think this is the more fruitful approach.
All: The Cayley Dickson generalizations discussed in wikipedia: R C H
O did present an "answer" in that there are successful numeric
extensions, that complex numbers "are not alone". As much as I wish
computer graphics had used them for their transformations rather than
4-tuples (homogeneous coordinates) and 4-matrices, I'm not sure just
how quaternions differ in theory from linear algebra, which simply
started in on generalized n-tuples.
In other words, simple n-tuple algebras might have put all these
generalizations from R into a single framework. Why *aren't* complex
numbers simply our first use of 2-tuples, unified with the rest of
linear algebra. Possibly the answer is that, yes linear algebras uses
n-tuples, but they focus on very different matters such as linear
independence, spanning sets, projections, subspaces, null spaces and
so on.
Fun! So now I hope I can find some interesting problems that ONLY can
be handled with some of these non linear algebraic higher number
systems. Interestingly enough, I believe all of the extensions
mentioned, as well as all of linear algebra, have the same cardinality
.. the continuum, right?
Thanks for the insights,
-- Owen
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FRIAM Applied Complexity Group listserv
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============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
