alternative algebra -- featuring the octonions! 

Yes but restricted associativity like alternativity and the Moufang laws are very interesting. Moufang loops are truly exotic symmetry objects that generalise the group concept, using a subtly "twisted" form of composition of invertible functions, that may even be applied to Moufang loop actions and Moufang loop representations, generalisations of group actions and group representations. Notice that the Jacobi identity for Lie algebras is a consequence of the associativity of the Lie group, and of the universal enveloping algebra. For analytic Moufang loops, the corresponding rule to alternativity is the Malcev identity of Malcev algebras.

I'd love too

I'd love too@@max.caimits

Yes! Construction please

And the diagram of the complex numbers (e1)

It contains the quaternion diagram 7 times in fact (e.g. edges like e1 -> e5 -> e6), unless I've miscounted. This is similar to how the quaternions themselves contain three copies of C. Presumably this generalizes with the 2^n system containing (2^n)-1 copies of the 2^(n-1) system.

@@Alex_Deam Yes, we can count them. Any two distinct imaginary octonions define a quaternionic subalgebra, and the third basis of the quaternionic subalgebra is uniquely implied. So you have 7 * 6 ways of getting the first two and 1 way of getting the third, but any permutation of the choices will give the same result up to sign. So as you guessed, 7 * 6 / 6 = 7 axis-aligned quaternionic subalgebras. This works after a fashion even if you don't restrict it to basis elements, they just can't be real scalar multiples of each other. But it doesn't work for the sedenions - the naive calculation 15*14*13 / 4! gives 113.75 octonionic subalgebras - because you also have to rule out picking the third one in the same quaternionic subalgebra and then quotient out 8 choose 3. So it looks like 15*14*12 / (8 choose 3) = 45 octonionic subalgebras of the sedenions, but it's complex and I may have missed something.

@@Tehom1 I think my (2^n)-1 guess was overhasty, your way of thinking of it in terms of choices and permutations seems like a better approach. However, I'm not sure I follow your naive calculation for the sedenions. I would've thought it would be (15!)/(9!7!). 15!/9! because you want to choose 6 sedenions (which presumably uniquely determines the 7th, and we want 7 because the quaternions has 7), and then divide by 7! to remove extraneous permutations. That works out to be 715, which is at least an integer so that's promising lol. In general, skipping some steps, it seems like the formula reduces down to 0.5*((2^n) choose ((2^(n-1))+1)) - and any binomial coefficient of the form (2^n) choose k with 0

@@Alex_Deam I see what it is. To define an octonionic subalgebra you can really only pick three imaginary basis elements freely up to independence. The others are implied by the choice. To see this, look at the Fano plane again and imagine you have picked - here I am running into the fact that I tend to use i,j,l as bases but Michael is using e_n - but say you have picked e1 and e5. That gives you e6 too on the same arrow if I am remembering the labels right. Then pick any of the others and it implies the rest of the basis because you have two points on every line and can calculate the third.

A more natural extension beyond the quaternions are the Clifford algebras. They are all associative, but they have zero divisors. The Cl(3,0) algebra, known as the geometric algebra, is super useful in physics.

To answer your question, any two imaginary octonionic bases define a quaternionic algebra, the third imaginary base is implied. It is their product up to choice of sign.

It's the same with the newspaper!

Get's?

... and what happens when you try double again to 32 :D

As much as I enthuse over the octonions, it gets boring after that. The sedenions have zero divisors, so a lot of equations become meaningless because if you are asking them whether the product of things involving any unknown can be zero, the answer is always yes. They also haven't got many interesting properties left to lose, though power associativity sticks forever I think.

(e[i]e[j])e[k] = -e[i](e[j]e[k]) only holds for some choices of i, j, and k. If you take i, j, k from the same "loop" in the diagram then you get associativity, e.g. (e[1]e[2])e[4] = e[1](e[2]e[4]). This means that the octonions have lots of associative subalgebras isomorphic to the quaternions, and in turn has lots of commutative associative subalgebras isomorphic to the complex numbers (and also in turn lots of subalgebras isomorphic to the real numbers)

They were invented to justify new RU-vid content.

The implication that things which are "just theoretical" are useless is an odd one. To recall an anecdote of Faraday's: "Before leaving this [...], I will point out its history, as an answer to those who are in the habit of saying to every new fact, 'What is its use?' [Benjamin] Franklin says to such, 'What is the use of an infant?'."

I know that octonions are used in string theory and shows up in physics related to supersymmetry

Sometimes it's multiply and divide by 1, but then, just as 0 is the "additive identity", 1 is the "multiplicative identity", so it's essentially the "0" for multiplication ....

Ordering

@@giorgiobarchiesi5003Yep If you’re wondering the last algebra in which you lose a property is the sedenions

@@the_linguist_ll I guess that when you loose all properties you don’t have an algebra any longer, right? But… dr. Furey was nevertheless able to obtain a Clifford algebra from the octonions, so there is still some hope, perhaps 😉

Yes I suppose so :D

I hardly know 'erions!

Yes. James Grimes (Numberphile) made a video a few years back where he pronounced it "sedonions" the whole time. People corrected him but I guess the pronunciation stuck.

וואלה לא ציפיתי למצוא מישהו שחושב שביבי סבבה פה (גם אני מעדיף אותו btw)

This is related to the fact that they aren’t associative. Complex numbers and Quaternions can be seen as a subalgebra of 2D and 3D Geometric Algebra. Specifically, geometric algebra has elements called bivectors which act to rotate vectors (which they also have) in a given plane. It turns out that the xy-plane bivector from 2D GA acts like i, while the xy, yz, zx bivectors from 3D GA act like the Quaternions i, j, k. Geometric algebras are defined such that they’re always associative. For this reason the Octonions are not a subalgebra of any Geometric Algebra! If you try to apply the same logic as for the previous algebras to 4D GA (we’re taking the “even subalgebra”) you get something that behaves like the split-biquaternions if I remember correctly. Geometric algebra is built with vector rotation in mind, and can extend to any number of vector dimensions. Given that it’s always associative, I believe that means none of the Cayley-Dickinson Algebras beyond the Quaternions can possibly describe rotations (at least not “cleanly”) To me, though, it makes them almost more exciting. What do they describe other than rotations? Why do the two derivations (GA and C-D) diverge? Or actually, why did they converge in the first place?

@@kikivoorburg I agree it's a fascinating mystery why they converge at all. I suspect it's a coincidence of "there are only so many algebras with so few moving parts" so to speak.

What do you mean?

Quaternions are used to smoothly describe 3d rotations, as in computer graphics. Pure imaginary unit quaternions double-cover 3d rotations, but this avoids gimbal lock and edge cases. No idea about octonions, though.

There are a few niche uses, for instance the action of the tensor product of the octonions, the quaternions, and the complex algebra on itself splits into a representation of spacetime (the poincare group) and a representation of the standard model of particle physics. Which is very neat, probably indicates something important, but isn't really a mainstream thing in physics.

@@Tehom1 Thank you! I would love to know more about this but, sadly, it would be beyond my understanding.

Look into quaternion algebras. Also highly recommend the book by Conway and the paper by John Baez.

The Cayley-Dickinson construction basically takes two copies of the previous algebra and combines them to get the next one.

@@waverod9275 interesting. Is it possible to construct algebras in an odd vector space, rather than even?

@@Efesus67 that I don't know. I know the simplest/obvious attempt (basis of 1 and two separate square roots of -1) at three doesn't work as a division algebra, because that's what Hamilton tried before he figured out quarternions. But that by itself doesn't rule out an algebra with three basis vectors. Maybe someone else here knows?

@@waverod9275 yeah, what about quintonions? Lol

Don't be daft. Words in mathematics, such as "alternating" and "alternative" have strict definitions. Let the Republicans show off their stupidity and ignorance every time they open their mouths, and get on with the interesting stuff (i.e. mathematics).