alternative algebra -- featuring the octonions!
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- Опубліковано 14 сер 2023
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Non associative algebra is such an unfamiliar field and it always surprised me how important associativity is to us
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 to see a video above Cayley-Dickson construction! As always thanks again!
I'd love too
I'd love too@@max.caimits
Yes! Construction please
16:50 what i find really cool about this diagram is that it contains the diagram of quaterions inside it (e1 - e2 - e4) which is it's subalgebra
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.
Speaking personally a Cayley-Dickinson Construction seems a lot sweeter than gamma reflection via double and contour integration so yes please! More on C-DC
14:31 lol, I also do that “= … =“ sometimes when I’m taking notes and there’s a long tedious calculation that isn’t very illuminating to my understanding and is just part of the process.
I also sometimes use “=> … =>” for the same reasons.
Professor Penn is a teacher in the class pedagogy hard to find elsewhere. His style of delivery hits the heart of motivated learners. Master of ab-initio teaching from basics of Number Theory.Can't go anywhere else to learn about happiness.
A series on Geometric (Clifford) Algebra would be great.
Associativity is one of those properties that you see everywhere but never question why its so important, wonderful video
I've been fascinated by quaternions for a long time, but octonions have always intimidated me. I appreciate the brief introduction to octonions given here!
And I think a video on the Cayley-Dickson construction would be interesting, too!
Newer developments in "Clifford-Algebra" is a gamechanger for maths.
It makes hard problems look simple. They should teach this in school.
Would be nice to see a brief discussion on sedenions and power-associativity.
... 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.
16:50 i started thinking the diagram would complete if it was embedded in a circle but then i realized i'd still be missing 3 loops
Definitely want to see the Cayley-Dickinson construction, especially the sedenions, the gradual loss of the usual properties of multiplication, and what, if any, uses there are (in wider mathematics) for the Cayley-Dickinson algebras of still larger dimensions.
yes please. the stack of increasing dimensional numbers has intrigued me for decades, ever since I learned to use quats for representing movement in R^3. I would also like to see how to instantiate octonions/sedonians, in order to embed quats, complex, &cet inside the higher-order algebras. IS it sufficient, for example to define i ,j, k by e_i ,e_j, and e_k where none of the subscripts are equal?
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.
Great video!! Would love to see connections between octonions and superstrings. And, yes, Cayley-Dickenson as well.
I’d love to see a video on the Cayley-Dickenson construction!
I'm interested in the Cayley-Dickson construction along with any more Octonian and Sedonian content.
This is the first time I've met them.
loved this video :) nice to see stuff outside undergrad
shhh shhh shhh, you had me at "alternative".
Hexagons may be the best agons, but octonions are the best onions.
With commutative properties, we usually call things anti-commutative if when swapping the two factors you pick up a minus sign, like how in the quaternions you have ij = k but ji = -k. WIth the octonions, it seems like (e[i]e[j])e[k] = -e[i](e[j]e[k]). Are there alternative algebras that aren't "anti-associative" like this?
(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)
Hi Dr. Penn!
Very cool!
Please make a video on the Cayley Dickinson Construction!
I never knew that the multiplication table for the octonions could be viewed as being embedded in a projective plane before. That's pretty cool!
It would be interesting to compare the Cayley-Dickson construction with the related Clifford algebras.. You can get Complex Numbers and Quaternions either way, but the Octonions aren’t a Clifford algebra _because_ of the non-associativity
If you start with Z2 (boolean algebra) and quotient with x(x+1)=1, you get F4. x(x+1)=1 means x and not x is true, in other words F4 is a multivalued logic where the two new elements are x and not x, and their product is true. 4-valued logics are a bit of a jump, because you lose monotonicity, but you can identify x and x+1, they are isomorphic, and the result is the 3-valued logic RM3, after a bit of extra algebra involving the tensor-hom adjunction. It's the same sort of progression as R => C => H => ... except D1, D2, D3, and D4 are the only lattices (D1 is trivial) that form interesting logics. Beyond that you lose more properties and it becomes more complex. But you can still make infinite valued logics. RM3 it should be noted is the famous "yes, no, maybe" logic that is taught in grade school (what, your school didn't teach that? Write them an email).
Was wondering where you were heading for most of video. Payoff was worth it. So very cool. More, please.
Can you make a video on A-calculus? Calculus over an arbitrary algebra.
Yes please 🙏
The diagram for multiplying octonion basis vectors is the 7-point projective plane [the pp over Z[2]?). Is there some deep reason for this? Or is it just that, for any pp:
i) two points determine a unique line
ii) two lines intersect in a point
iii) there are at least 3 points on a line [exactly 3 points in the 7-point plane]
Thus any two basis vectors can be "joined" by a line [multiplied] and yield a unique product [the 3rd point on the line]
Note: One of the "lines" in the 7-point plane is a circle because Z[2] can't be embedded in R, is not a sub-field of R - something like that
Which is more popular, adding zero or multiplying by one?
you should do the sedonions next!
19:18
What I find fascinating about octonians is that they don't seem to represent rotation in some N-dimensional space the way complex numbers and quaternions do. It seems like such a pattern, with C representing 2D rotation and H representing 3D. I don't know about sedonians and 6D; it seems like they would work but a bit of a mess to try to prove, one way or the other.
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.
I swear in every video the writing becomes smaller. Soon it will be downright microscopic!
It's the same with the newspaper!
Get's?
16:50 i do not get the logic behinf the ordering of e1,...,e7 in the diagram :/
When Michael did a piece on the quaternions I opined that they were not useful in modern mathematics and was told this wasn't so. I'd love to hear where they are used, and I'd also love to hear from anyone who has used octonions!
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.
Hmmmm why do the dimensions satisfy 2^n where n is the natural numbers and 0. - nevermind saw the final part at end - guess will be gladly anticipating when that's explained with the Cayley Dickson Construction.
so geometrically, you can use Dual Quaternions to projectively describe rotations and translations in R³ in a unified way.
Clearly, that is *not* equivalent to Octonions. It does, however, have the same number of dimensions. And the units of Octonions are described by the Fano plane which is the smallest possible projective plane (the projective plane of just seven elements)
That's projective in a very very different way than the Dual Quaternions.
But either way, I have to wonder:
Do the Octonions in any way shape or form describe interesting transformations in R³? How?
Obviously, by simply taking an approrpiate subset of them, you can use half-Octonions to do Rotations in R³ in the same way as you could use Quaternions.
But what would the remaining half do? In the case of Dual Quaternions, that's where you'd get Translation, afaik.
Does that mean flexible algebras have an "alternator"?
Yes I suppose so :D
I think that octonions are underused. It is not hard to write down a matrix representation for them (I believe) under the assumption that one is always associating, say right to left.
With real data, one typically only associates this way: ie O3(O2(O1(Data))), where O3 is an operation on the data. So, I don't really understand why we neglect so much non associative operations. Any comments?
Thank you for the lecture. Yes we do want to see the construction of Octonions and more. Objects made of numbers and operations between them define an algorithm. We should try to find a more general algorithm that generates all such objects made of sets of objects and operations and automate inferences of theorems from the rules. However, there is a problem. Mathematics is the direct result of our structure as neural networks and especially neural networks which are capable of language, e.g. in engineering, Transformer Neural Networks. Mathematics is a minimal language which is based on simple axioms. Better understanding of neural networks will automate mathematics. Understanding the limits of neural networks will result in understanding the limits of mathematics, such as Kurt Godel's Incompleteness Theorem. Recommended for reading is Neural networks and analog computation beyond the Turing limit by Hava T. Siegelmann. My conjecture is that if physical neural networks can represent recursive numbers, they can know if a theorem is correct or not even if no theorem can be derived from the axioms. This conjecture needs to be validated.
14.36
It should be a1(a2a3) instead of a1(a2a2)
"Sedenions" 😇
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.
I want to see the 32 dimensional Calyey-dickenson algebra past the sedonians :)
So the entire math is impossible to do without adding and subtracting some version of zero 😂
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 ....
Is it sensible to think of commutators if you can get a numerical value for them as a "measure" of how commutative your algebra is? - same with your associators that you show here? I'm thinking of numeric values to interpret would be the integers so far - not sure how well you can interpret the reals then again I hesitate because my real analysis is a bit weak. Otherwise are they only good for string symbol code algebra manipulations?
Has this alternative algebra any use or is it just theoretical?
They were invented to justify new UA-cam 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
Why do they increase by powers of 2?
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
DISTRIBUTIVE LAW HAS A FLAW ?
What do you mean?
How do Alt Algebras and Lie Algebras relate to each other? Lie Alg's have that 3-way translation sum zero property, I forget its proper name. It's not "too" dissimilar from the alternative zero equations.
a times bb (Bibi??)...lol. A controversial prime minister of Israel, Benjamin Netanyahu...
Actually I like him, and also this theme of special algebras
וואלה לא ציפיתי למצוא מישהו שחושב שביבי סבבה פה (גם אני מעדיף אותו btw)
Please don't call it "alternative". That word has been taken by certain Republicans.
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).