[HN Gopher] The exceptional Jordan algebra (2020)
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The exceptional Jordan algebra (2020)
Author : gone35
Score : 103 points
Date : 2025-03-17 07:30 UTC (15 hours ago)
(HTM) web link (cp4space.hatsya.com)
(TXT) w3m dump (cp4space.hatsya.com)
| aap_ wrote:
| What i think is super cool about h3(O) is that you have scalars,
| a vector and two spinors in one object with the jordan-product
| giving you all the products between them that have to do with
| triality. See section 3.4 of John Baez' paper on the octonions.
| bollu wrote:
| How does one formally define a spinor? I've seen the definition
| of a spinor field as "things that transform like a spinor", and
| a spinor as a "representation of the spin group" (which
| representation), but I would like to know a canonical mathsy
| definition of what the heck a "spinor" is! May I please have
| one? :)
| dapperdrake wrote:
| How come "representation of the spin group" is an
| insufficient starting point?
|
| Spin group seems like they either have a specific group (from
| Algebra) in mind or that spins are at least defined by
| choosing a specific group (a set with a binary operation
| satisfying the group axioms/definition).
|
| A "representation" also has a definition in Algebra with
| regards to groups. There are group homo-morphisms between two
| groups. This means you have a mapping that preserves group
| structure. Representation theory is about mapping groups into
| the set of matrices or a subset of matrices "with numbers in
| the matrices." Then there are group actions (don't care for
| the name) - basically/conceptually a set of functions that
| behaves like a specific group under composition, but way more
| notation around that. Finally, category theory looks at
| "groups of groups" with the binary operation being homo-
| morphisms between the "inside/smaller/contained/internal"
| groups thus forming a larger "outside" group called a
| category. Because this involves talking about sets of sets
| you end up also needing the term "class" from set-theory.
| JadeNB wrote:
| It's not that "representation of the spin group" is
| undefined, but that there are too many of them for it to
| pin things down uniquely. (In this case, fortunately, it's
| not hard to say which representation it is (see
| https://news.ycombinator.com/item?id=43388052), but just
| saying "a representation" isn't enough.)
| whatshisface wrote:
| While we're talking about representations, there's
| something I've always wondered. Why are the objects that
| the maps which are the representations act on also called
| representations? Spinors don't act as the spinor group, S
| [?] Hom(Spinor,Spinor) does.
| aap_ wrote:
| Unfortunately there are slightly different but related
| notions of what spinors are. One key idea is indeed how they
| transform. A spinor ps transforms with a transformation S [?]
| Spin(n) in a one-sided way: ps -> Sps. A vector v in contrast
| transforms in a sandwich way (with the inverse on one side):
| SvS^-1. Intuitively this explains why spinors transform "half
| as much" as vectors, e.g. the 720deg vs 360deg rotational
| symmetry that shows up in physics. So for any S [?] Spin(n)
| the sandwich-action (S|S^-1) gives you the corresponding
| element of SO(n). Because a negative sign on S squares away
| in that case, S and -S map to the same SO(n) action, and
| therefore Spin(n) is said to be the double-cover of SO(n)
| (personally I think it would be better terminology to call
| SO(n) the half-cover of Spin(n)).
|
| So a spinor could be said to be anything whose symmetries are
| a Spin group. Spin groups are easily constructed in clifford
| algebras and it turns out that they have matrix
| representations. Whenever you have matrices (linear maps) you
| may wonder what the vector space is that they act on (i'm now
| using the term "vector" abstractly, not in contrast to
| spinors as above). Well, those are the spinors (technically
| pinors)! Another definition of spinors is that they live in a
| minimal left ideal of a clifford algebra. This does not sound
| very intuitive at first, but it can be understood easily as
| simply taking the matrices with only one non-zero column.
| These are really not very different from colunm vectors then.
| There seems to be some confusion about pinors and spinors in
| that perspective though...it just seems to be a somewhat
| confusing concept in general.
|
| The spinors/vectors relevant to the article are those of
| Spin(8), which has something to do with triality (still need
| to understand all of this better myself). The basic idea is
| that in Cl(8) the vectors and spinors are both 8-dimensional
| and the algebra can be generated by left-multiplication of
| octonions. So there are some interesting symmetries
| occurring. The Baez-article goes into that too but it could
| have been a bit more explicit for my taste.
|
| I hope some of that made sense, i don't know your background.
| I'm still trying to wrap my head around this topic myself and
| have been for about 2 years now.
|
| Maybe check out the "spinors for beginners" series on
| youtube. It's very good and quite extensive.
| JadeNB wrote:
| > The spinors/vectors relevant to the article are those of
| Spin(8), which has something to do with triality (still
| need to understand all of this better myself).
|
| Spin(8) itself doesn't have much to do with triality; it's
| just that triality describes an unusual symmetry among
| representations of Spin(8), due to an unusual outer
| automorphism. (Of course, from some perspectives, that
| means that Spin(8) has _everything_ to do with triality,
| but I hope my meaning will be clear.) The best accessible
| mathematical explanation of triality I know is from Baez:
| https://math.ucr.edu/home/baez/octonions/node7.html.
|
| > So a spinor could be said to be anything whose symmetries
| are a Spin group. Spin groups are easily constructed in
| clifford algebras and it turns out that they have matrix
| representations. Whenever you have matrices (linear maps)
| you may wonder what the vector space is that they act on
| (i'm now using the term "vector" abstractly, not in
| contrast to spinors as above). Well, those are the spinors
| (technically pinors)!
|
| One has to be a little careful here, because algebras have
| _lots_ of representations, and there 's no one
| representation that _a priori_ may be said to be " _the_
| vector space on which they act " ("the" rather than "a").
| For example--though it's a bad example because it's not a
| faithful action--Spin(8) naturally double covers SO(8), but
| we don't want to take the resulting 8-dimensional
| orthogonal representation (the "vector representation").
| Instead, we want to take one of the three fundamental
| representations permuted by the triality automorphism (the
| V_1, V_2, V_3 in Baez's article).
| aap_ wrote:
| I slightly edited my post while you were writing yours.
| I'm aware of the article but I still think it could be
| clearer. I'm a big fan of constructive proofs but that
| does not seem the case for most mathematicians, so i find
| a lot of stuff in math very unsatisfying and have to
| rediscover it myself via different routes. Educational
| for sure but not always so easy.
|
| And yes, of course there are many possible matrix reps,
| sorry i was not being precise. The ones i was referring
| to are the ones given here: https://en.wikipedia.org/wiki
| /Classification_of_Clifford_alg...
| JadeNB wrote:
| > How does one formally define a spinor? I've seen the
| definition of a spinor field as "things that transform like a
| spinor", and a spinor as a "representation of the spin group"
| (which representation), but I would like to know a canonical
| mathsy definition of what the heck a "spinor" is! May I
| please have one? :)
|
| For Spin(8), three of the four fundamental representations
| are conjugate, and so we can use any one of them to define
| spinors.
| Tewboo wrote:
| Intriguing! I remember when Jordan algebras were the talk of the
| theoretical math town. Time to revisit the scene!
| mjreacher wrote:
| I wonder if there is any connection of this being on the front
| page today with this paper [0] being uploaded on arXiv today or
| if it's just pure coincidence.
|
| [0]: https://arxiv.org/abs/2503.10744
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