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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