For the discussion of math. Duh.
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dhokarena56 wrote:I know I'm fascinated by quaternions and octonions and such. Here's my question: does an algebra system exist for any dimension 2^n where n is a positive integer?
You can create "an algebra system" in any dimension -- the question is just what rules do and do not break in that dimension.
What's nice about quaternions and octonions (and real/complex numbers) is they are Division algebras, a property that can *not* be preserved in higher dimensions (over the real numbers, anyway). However, if you're just interested in higher-dimensional analogues of octonions and you don't really care about the division ring property so much, then I present to you: Sedenions (and in more generality, the Cayley-Dickson construction).
NathanielJ wrote:What's nice about quaternions and octonions (and real/complex numbers) is they are Division algebras.
I am a little bit disturbed by that Wikipedia article. When I was in graduate school anything called an 'algebra' was automatically assumed to be a ring, i.e. both associative and to have a multiplicative identity (=1). Care was taken to mention that in analysis (like some function algebras) the assumption about the existence of 1 is dropped (Jacobson calls such beasts 'rngs' to emphasize the absence of '1'). Similarly, at the beginning of a Lie algebra course it was noted that these are non-associative. IOW there were algebras and non-associative algebras. That wikipedia article leaves you with the impression that there are algebras and associative algebras. Surely in the context of algebra associativity of multiplication is the norm, and a deviation from that standard is the one in need of extra specification.
Also, why does any brief description of division algebras stick to the ever so limited theory of 2 extensions of the reals, and ignores the infinitely many extensions of other number fields? Is the connection to topology the reason?
Jyrki wrote: Surely in the context of algebra associativity of multiplication is the norm, and a deviation from that standard is the one in need of extra specification.
Yeah, associativity is really nice but if you are mucking around in interesting spaces why on Earth would you not mention all the algebraic properties you can use? Maybe not all the time, but at least once...
As for "the norm", Anti-commutative Lie Algebras are non-associative, and they aren't uncommon.
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