Self-Referential Algebraic Structures

July 8, 2020

Self-Referential Algebraic Structures

Posted by John Baez

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Any group acts as automorphisms of itself, by conjugation. If we differentiate this idea, we get that any Lie algebra acts as derivations of itself. We can then enhance this in various ways: for example a Poisson algebra is both a Lie algebra and a commutative algebra, such that any element acts as derivations of both these structures.

Why do I care?

In my paper on Noether’s theorem I got excited by how physics uses structures where each element acts to generate a one-parameter group of automorphisms of that structure. I proved a super-general version of Noether’s theorem based on this idea. It’s Theorem 8, in case you’re curious.

But the purest expression of the idea of a “structure where each element acts as an automorphism of that structure” is the concept of “rack”.

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Even simpler than a rack is a “shelf”.

Alissa Crans defined a shelf to be a set $S$ equipped with a binary operation $\triangleright \colon S \times S \to S$ such that each element $a \in S$ gives a map $a \triangleright - \colon S \to S$ that is a shelf endomorphism.

If you don’t like the circularity of this definition (which is the whole point), what I’m saying is that a shelf is a set $S$ equipped with a binary operation $\triangleright \colon S \times S \to S$ such that

$a \triangleright (b \triangleright c) = (a \triangleright b) \triangleright (a \triangleright c)$

This is called the self-distributive law, since it says the operation $\triangleright$ distributes over itshelf. (Sorry: “itself”.)

Actually this is a left shelf; there is also a thing called a “right shelf”. Any group is both a left shelf and a right shelf in a natural way, via conjugation.

We can similarly define a rack to be a set $S$ equipped with a binary operation $\triangleright \colon S \times S \to S$ such that each element $a \in S$ gives a map $a \triangleright - \colon S \to S$ that is a rack automorphism.

In other words, a rack is a shelf $S$ where each map $a \triangleright - \colon S \to S$ is invertible.

It’s common to write the inverse of the map $a \triangleright -$ as $- \triangleleft a$ . Then we can give a completely equational definition of a rack! It’s a set $S$ with two binary operations $\triangleright, \triangleleft \colon S \times S \to S$ obeying these identities:

$(c \triangleright b) \triangleright a = (c \triangleright a) \triangleright (b \triangleright a), \qquad a \triangleleft(b \triangleleft c) = (a \triangleleft b) \triangleleft(a \triangleleft c)$

$(a \triangleleft b) \triangleright a = b , \qquad a \triangleleft(b \triangleright a) = b$

These axioms are slightly redundant, but nicely symmetrical.

If we have a shelf that’s a smooth manifold, and the operation $\triangleright$ is smooth, and an element $e \in S$ obeying $e \triangleright a = a$ for all $a \in S$ , then I think we can differentiate the shelf operation at $e$ and get a “Leibniz algebra”.

A Leibniz algebra is a vector space $L$ with a bilinear operation $[-,-] \colon L \times L \to L$ such that each element $a \in L$ gives a map $[a,-] \colon L \to L$ that is a Leibniz algebra derivation. In other words, the Jacobi identity holds:

$[a, [b,c]] = [[a,b], c] + [b, [a,c]]$

Again, this is really a left Leibniz algebra, and we could also consider “right” Leibniz algebras. Any Lie algebra is naturally both a left and a right Leibniz algebra.

I’ve been trying to understand Jordan algebras, and here’s one thing that intrigues me: in a Jordan algebra $A$ , any pair of elements $a,b \in A$ acts to give a derivation of the Jordan algebra, say $D_{a,b} \colon A \to A$ , as follows:

$D_{a,b} (c) = a \circ (b \circ c) - b \circ (a \circ c)$

If $L_a$ means left multiplication by $a$ , this says $D_{a,b} = L_a L_b - L_b L_a$ .

This property of Jordan algebras seems nicer to me than the actual definition of Jordan algebra! So we could imagine turning it into a definition. Say a pre-Jordan algebra is a vector space $A$ equipped with a bilinear map $\circ : A \times A \to A$ such that for any $a,b \in A$ , the map $D_{a,b} \colon A \to A$ defined above is a derivation, meaning

$D_{a,b}(c \circ d) = D_{a,b} (c) \circ d + c \circ D_{a,b}(d)$

for all $c, d \in A$ .

Puzzle. Are there pre-Jordan algebras that aren’t Jordan algebras? Are any of them interesting?

Posted at July 8, 2020 7:28 PM UTC

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Re: Self-Referential Algebraic Structures

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I don’t know about the puzzle, but self-actions have also been on my mind recently, so here are some random thoughts somewhat “off the shelf”. Let’s start with the definition of shelf:

A shelf is a set $S$ equipped with a binary operation $\triangleright \colon S \times S \to S$ such that each element $a \in S$ gives a map $a \triangleright - \colon S \to S$ that is a shelf automorphism.

If you don’t like the circularity of this definition (which is the whole point), what I’m saying is that a shelf is a set $S$ equipped with a binary operation $\triangleright \colon S \times S \to S$ such that

$a \triangleright (b \triangleright c) = (a \triangleright b) \triangleright (a \triangleright c)$

Are you sure that this self-distributivity equation implies that every self-map $a \triangleright -$ is invertible? For example, it seems to me that the implication also makes every Heyting algebra into a shelf, because both sides then are equivalent to $(a \wedge b) \Rightarrow c$ , but clearly the map $a \Rightarrow -$ is almost never invertible. So unless I’ve made a silly mistake, already implication on the Booleans defines a shelf, and in general the elements of a shelf only acts by endomorphisms rather than automorphisms. But building in inverses and hence invertibility shouldn’t be difficult.

Here are some further questions on the category theory of shelves:

Puzzle 2: What are the elements of the free shelf generated by one element? The free shelf generated by $n$ elements?

Puzzle 3: Does the category of shelves have any extra structure which encodes the self-actions?

Puzzle 4: If so, does the category of shelves have some kind of universal property, e.g. something like being terminal among all categories of self-acting objects?

Perhaps Alissa has already looked into questions of this sort? I haven’t read the whole thesis yet.

I’m not familiar enough with the details to say much about this, but shelves also remind me of Wolfgang Rump’s cycloids and L-algebras. A cycloid is a set with a binary operation satisfying the equation $(a \triangleright b) \triangleright (a \triangleright c) = (b \triangleright a) \triangleright (b \triangleright c) ,$ and an L-algebra is essentially a cycloid together with a (suitably defined) unit element. I bet that the appearance of self-similarity and the Yang-Baxter equation in both Alissa’s thesis and Rump’s papers is no coincidence. Again implication in a Heyting algebra defines an L-algebra structure, and there also seem to be connections with quantum logic!

Posted by: Tobias Fritz on July 8, 2020 10:58 PM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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Tobias wrote:

Are you sure that this self-distributivity equation implies that every self-map $a \triangleright -$ is invertible?

No, I’m the opposite of sure that this is true. It’s false. So, I’m going to correct my post a bit.

People often consider racks, which are shelves where each operation $a \triangleright -$ has an inverse. We can call this inverse $- \triangleleft a$ , and then we can give an equational presentation of the theory of racks (see the link).

Posted by: John Baez on July 8, 2020 11:10 PM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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Oh, neat! I had seen the definition of rack given in terms of the two binary operations, but I didn’t know that it’s equivalent to shelf plus invertibility. I’ll have to think more about this.

By the way, do you have any intuition on why physical observables should have such self-actions, other than the quasi-empirical observation that this seems to be the case in all physical theories? This is the kind of question that may not have a good answer, but if it does then it could be very interesting.

Posted by: Tobias Fritz on July 8, 2020 11:18 PM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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Tobias wrote:

By the way, do you have any intuition on why physical observables should have such self-actions, other than the quasi-empirical observation that this seems to be the case in all physical theories? This is the kind of question that may not have a good answer, but if it does then it could be very interesting.

So far I only have a vague philosophical answer, where I’m using the word “philosophical” in the somewhat derogatory sense that tends to annoy actual philosophers. It’s this:

The usual concept of “observable” treats observation as something an external “observer” can do to the system, so it’s rather hard to completely formalize unless we treat the system as an open system and describe the observer too. The idea that a system is described by quantities each of which gives rise to automorphisms of that system is somehow more “self-contained”, and thus it may allow us to sidestep some problems with formalizing the concept of “observable”.

I would like to develop this further.

In particular, here’s a dumb question. I seem to be saying that “observables” can only be deeply understood using something like a monoidal category of systems. The categorical quantum mechanics crowd loves to work with monoidal categories of systems. Do these people have a good framework for describing “observation” or “measurement” processes using monoidal categories—one that can actually handle real-world observations, or at least some simplified version of them? A really good treatment should explain why we use “observable” to mean “self-adjoint operator” (or normal operator).

I feel I should know this.

Posted by: John Baez on July 8, 2020 11:32 PM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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Very nice! So from that perspective, the self-actions are closely related to open systems. Would you say that for actual physical systems, you would therefore expect the map from observables to automorphisms (or 1-parameter groups thereof) should be injective? Since whenever it’s not, then there will be different observables that look the same to the observer, who will therefore model the system in a way that identifies them?

Personally I’m a bit skeptical about whether monoidal categories are a sufficiently general framework for this idea: it’s us decomposing the universe into perceived subsystems rather than the universe composing the systems for us. But at least one can get a large range of examples.

Posted by: Tobias Fritz on July 8, 2020 11:50 PM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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Actually this almost sounds like you’re defining an observable to be an automorphism of the total system (containing a measurement apparatus).

Ignoring the different between measured system and total system for the moment, it seems that we then have the automorphism group as the fundamental structure from which the shelf or Leibniz algebra structure is derived. Hmm.

Posted by: Tobias Fritz on July 9, 2020 12:02 AM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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Tobias wrote:

So from that perspective, the self-actions are closely related to open systems.

Hmm, I’d thought of self-actions as a way of sidestepping the need to think about open systems and “observation”, and thinking of a physical theory as more of a self-contained thing. But maybe this is just the other side of the same coin. I don’t understand much about this….

Would you say that for actual physical systems, you would therefore expect the map from observables to automorphisms (or 1-parameter groups thereof) should be injective?

That’s an interesting question! For C $^\ast$ -algebras this is saying the C $^\ast$ -algebra has no center. Elements of the center give superselection rules. Of course we usually allow the center to include $\mathbb{C}$ , and self-adjoints in here generate one-parameter groups of phases $\exp(i \theta)$ . These act trivially on observables (by conjugation), but they’re still somehow fundamental to quantum mechanics.

So, I can’t quite answer your question, but I’ll just say that observables that give trivial one-parameter groups of automorphisms of observables are very special. They include “constants”, which are very weird as observables go: they’re the observables you don’t actually need to look at your system to measure!

Posted by: John Baez on July 9, 2020 12:06 AM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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Ugh, sorry, I had misunderstood your idea! It’s more like the opposite of what I thought it was… Sometimes reading twice isn’t enough.

Oh, right, superselection sectors are a great example to look at. Naively I would think that e.g. electric charge deserves to be considered a nontrivial observable, as there seem to be obvious ways to measure it. So then I would tentatively answer my own question with “No, the map from observables to automorphism does not need to be injective”, even when considering observables modulo constants.

On the other hand, superselection charges generate global gauge symmetries, and this makes the tentative conclusion of the previous paragraph conflict with my view that gauge symmetries aren’t really any different than “physical” symmetries. Plenty of food for thought. Thanks!

Posted by: Tobias Fritz on July 9, 2020 12:35 AM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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Tobias wrote:

Oh, right, superselection sectors are a great example to look at. Naively I would think that e.g. electric charge deserves to be considered a nontrivial observable, as there seem to be obvious ways to measure it.

The electrical charge of the whole universe is a quantity that commutes with all others, and this defines a superselection sector. It’s not extremely easy to measure. The electrical charge in a compact region of space is a lot easier to measure (you can use Gauss’ law), but it doesn’t commute with all other observables.

This has some funny connection with the “open systems” theme, since the compact region of space is like an open system.

Posted by: John Baez on July 9, 2020 2:41 AM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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John wrote:

The categorical quantum mechanics crowd loves to work with monoidal categories of systems. Do these people have a good framework for describing “observation” or “measurement” processes using monoidal categories—one that can actually handle real-world observations, or at least some simplified version of them? A really good treatment should explain why we use “observable” to mean “self-adjoint operator” (or normal operator).

To me, it seems like the definition of “observable” as “self-adjoint operator” will only make sense as a limiting case of something more general. Here’s a quote by John Bell, writing to Rudolf Peierls, that maybe expresses why:

I have the impression, as I write this, that a moment ago I heard the bell of the tea trolley. But I am not sure because I was concentrating on what I was writing. Maybe I really heard it. But if so it was faint, indicating that the trolley is still distant. The strength of the sound was a ‘pointer’ ‘measuring’ the distance of the trolley. Other ‘pointers’ will be people passing my open door towards the trolley when it finally reaches the nearby corner. But they might be going to a lecture. This sort of thing is going on more or less all the time more or less everywhere. The clouds in the sky are ‘pointer’ to humidity. As I write I am aware of a small portion of the sky, but only in the tail of my eye. I could go to the window and look more definitely. Probably someone not far away is gazing through the window. More likely he is daydreaming than making meteorological observations. But ‘Measurement’ more or less. From this to the Stern–Gerlach experiment, it seems to me, is only a matter of degree. As I remember the output of the original experiment, it looked something like […] Two more or less distinct crescents. A more modern experiment will surely do better, but residual gas scattering, if nothing else will give residual confusion. Remembering also that the silver grains have finite size, and take a finite time to form, it seems clear to me that the ideal instantaneous experiments of the text-books are not precisely realised anywhere anytime, and more or less realized more or less all the time more or less everywhere. I think then the status of such ‘measurements’ should be like that of the ideal reversible heat engines of old fashioned thermodynamics — with an honourable place in phenomenology, but no place in fundamental theory.

When I was a student, I had to take a lab course, and one of the labs we did was a quantum computing experiment, attempting to implement the Deutch–Jozsa algorithm on a little vial of $^{13}CHCl_3$ inside a great big dewar surrounded by all kinds of wires. Reading out the result of a computation was quite a job, with sending in a sequence of RF pulses, recording resonance signals from a superheterodyne coil, taking a Fourier transform, comparing the result to a frequency spectrum recorded under different circumstances to understand the thermal background, adjusting this and that in the software to minimize the imaginary part of data that we’d really like to be real-valued… Quite a lot of idealization and background theorizing goes into being willing to call that whole affair the evaluation of

$\langle 00 | \rho | 00 \rangle.$

It worked — I graduated — but it does rather underline the point that much of the empirical ground we have for trusting the theory comes from applying it to observations that are pretty amazingly approximate. (A theory that only yields useful predictions when everything is perfectly aligned is a pretty useless theory!)

A while prior to that lab class, I had the extraordinarily lucky experience of being able to look through a viewing port in a cryogenic cooling machine and seeing the superfluid helium transition. Nowadays, whenever I make my quarantine meals, I can turn on the electric stove and watch a blackbody spectrum shift its peak up from the infrared. Presumably, John Bell would call all of these ‘measurements’ (in quotation marks). My very open-ended question is whether the formalism of monoidal categories can make sense of the kind of precision continuum that he envisaged, from listening for the tea trolley through witnessing the quieting of turbulent helium and beyond.

Posted by: Blake Stacey on July 9, 2020 5:48 AM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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Blake wrote:

My very open-ended question is whether the formalism of monoidal categories can make sense of the kind of precision continuum that he envisaged, from listening for the tea trolley through witnessing the quieting of turbulent helium and beyond.

That seems a lot harder than my question about whether monoidal categories can say something interesting about why self-adjoint (or normal) operators are a good way to formalize observables in quantum mechanics.

I’m more of a mathematical physicist with some pretensions to theoretical physics than a philosopher of physics. So I don’t want to get into the complications John Bell is talking about: they’re important, but I don’t see how they’ll help me make progress in coming up with new theories of physics, or improving the mathematics of the theories we have. Maybe they would in the long run if I could fight my way through them! But it feels like a long march through ever deepening mud. I like things that are mathematically beautiful, shiny and elegant.

I think a monoidal category approach to some highly idealized and abstracted measurement setups could be shiny and elegant.

Posted by: John Baez on July 9, 2020 6:43 AM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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I guess I’m holding out a small hope that mathematical elegance might arise from the muck. I’m thinking of Benford’s law, which is a nice bit of math, and which originated when Newcomb noticed that some pages of logarithm tables were literally more grimy than others. Or, when studying how imperfect optics will blur a sharp image into a fuzzier one, we get into convolution, and how the Fourier transform turns convolution into multiplication. To me, that’s fairly elegant. (Learning it certainly made a lot of arbitrary-sounding trickery from introductory differential equations much more intelligible in retrospect.) Similarly, a hierarchy of probabilistic measurements of decreasing precision could be a setting where characterizing Shannon entropy in terms of information loss becomes relevant.

But that’s all quite vague, and now I need to think about whether any of the things I’m curious to say about convex cones can benefit from the language of shelves and racks and quandles.

Posted by: Blake Stacey on July 11, 2020 8:56 PM | Permalink | Reply to this

Alexander

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It seems like any Lie algebra is trivially a pre-Jordan algebra. Am I missing something?

Posted by: Alexander Shamov on July 9, 2020 10:55 AM | Permalink | Reply to this

Re: Alexander

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You’re right! So okay, both Lie algebras and Jordan algebras give what I’m (temporarily) calling pre-Jordan algebras.

Posted by: John Baez on July 10, 2020 12:01 AM | Permalink | Reply to this

Re: Alexander

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…what I’m (temporarily) calling pre-Jordan algebras.

The term’s already being used.

Posted by: David Corfield on July 10, 2020 7:45 AM | Permalink | Reply to this

Re: Alexander

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Thanks! Reading that reminded me that a “pre-Lie algebra” is a vector space with a bilinear operation $\triangleright : L \times L \to L$ obeying a certain identity that implies the bracket

$[a,b] = a \triangleright b - b \triangleright a$

makes $L$ into a Lie algebra. (I forget if this is implication is an iff.) This seems closely connected to my ‘pre-Jordan’ idea of imposing an identity that forces

$a \triangleright (b \triangleright -) - b \triangleright (a \triangleright -)$

to be a derivation, since in any Lie algebra

$[a, -]$

is a derivation, so by the Jacobi identity

$[a, [b, -]] - [b, [a, -]]$

is a derivation.

I’m getting a bit confused by all these similar ideas. But anyway, ‘pre-Jordan’ is not the right name for my idea!

Posted by: John Baez on July 10, 2020 7:39 PM | Permalink | Reply to this

Re: Alexander

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It’s not an iff. The commutator of any commutative algebra trivially gives a Lie algebra structure, but one can check that a commutative pre-Lie algebra is associative. So any commutative nonassociative algebra gives a counterexample.

However, what is true is that pre-Lie algebras can be characterized by the fact that their commutator bracket gives a Lie algebra structure such that the action of the algebra on itself by multiplication is a Lie algebra representation.

Posted by: Nick Olson-Harris on July 11, 2020 5:04 PM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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Did you seen the very recent

François Renaud, Higher coverings of racks and quandles – Part I ?

Posted by: David Corfield on July 9, 2020 8:23 PM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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No, I don’t keep track of most arXiv papers… interested in too many subjects. Thanks.

Posted by: John Baez on July 10, 2020 12:02 AM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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Has anyone defined an $\infty$ -rack? Is it also a space equipped with a binary operation $\rhd$ such that each $a\rhd -$ is an $\infty$ -rack automorphism? In that case the “circular” definition would presumably have to be interpreted by some kind of coinduction, since “being an $\infty$ -rack automorphism” would consist of extra data, namely an isomorphism $a\rhd (b\rhd c) \cong (a\rhd b) \rhd (a\rhd c)$ , which would then have to be respected by $a\rhd -$ in order for it to be considered a rack automorphism, giving a coherence diagram, which would commute up to extra data, etc. etc.

Posted by: Mike Shulman on July 10, 2020 3:23 PM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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It would be very nice to study $\infty$ -racks, but I mainly know about the study of higher quandles. A quandle is a rack obeying $a \triangleright a = a$ . (Every group becomes not just a rack but also a quandle under conjugation.)

Since shelves, racks and quandles are algebras of Lawvere theories, shouldn’t there be an automatic way to $(\infty,1)$ -categorify them? For example the work of Badziokh and Bergner lets us define homotopy algebras of Lawvere theories and prove they’re all equivalent to strict algebras in the category of simplicial sets.

Is there anything lacking in this approach?

By the way, my student Laurel Langford worked with Carter and Saito on quandle cohomology, and my student Alissa Crans worked with Przyticki on the homology of associative shelves. So, there seems to be an eerie rule at work in the universe saying that all my female students (and none of the male ones) work on the homological algebra of self-distributive structures. Thanks to the layer-cake philosophy of cohomology, this should be intimately connected to $(\infty,1)$ -categorified self-distributive structures, but I don’t think this been spelled out anywhere.

By the way: the idea of an associative shelf — a set with an associative self-distributive binary operation — sounds a bit weird. But it turns out any unital shelf (that is, one with a two-sided unit) is associative! And in fact a unital shelf is the same thing as a graphic monoid. Lawvere initiated the study of graphic monoids, and a commutative graphic monoid is the same as a semilattice. So, semilattices give a vast family of associative shelves, most of which aren’t racks.

Posted by: John Baez on July 10, 2020 7:27 PM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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That seems like a reasonable approach. But can we then say whether this definition coincides with the coinductive one?

Posted by: Mike Shulman on July 11, 2020 7:14 PM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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Maybe someone can try to prove that. Too hard for me!

Posted by: John Baez on July 11, 2020 7:32 PM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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This takes me back to the 4th Mile High conference in Denver. I’m grateful that John Huerta invited me.

Concerning Jordan algebras and proto-Jordan algebras, in my Denver talk I discussed a type of proto-Jordan algebra (in degree three case), which I had rediscovered after Vinberg, called T-algebras (where T is for ternary). I had stumbled across these algebras while generalizing the exceptional Lie algebras (in a finite-dimensional manner) and their Yang-Mills friendly gradings. Through further discussions with Garrett Lisi (at a UCLA conference) we also found a correspondence with Kac-Moody and Borcherds algebras.

Since Denver, we cleaned up the work on these T-algebras and defined HT-algebras, where H is for Hermitian. Such HT-algebras admit Peirce decompositions, as the exceptional Jordan algebra does with spin factor and spinor parts. This allows one to study (matrix) string theory in D=25+1, with 4096 spinor, as required for a deeper study of the Monster.

You may find these HT-algebras of interest. See:

arXiv:1910.07914 (math.RT)

An upcoming paper on the Monster and D=26+1 M-theory will be on arxiv within a week. We have found evidence for a D=26+1 supergravity and constructed its Lagrangian.

Posted by: Metatron on July 18, 2020 4:53 PM | Permalink | Reply to this

Re: Self-Referential Algebraic Structures

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I gather this is the paper you meant:

Alessio Marrani, Michael Rios, David Chester, Monstrous M-theory, arXiv:2008.06742

Posted by: David Roberts on August 19, 2020 1:46 AM | Permalink | Reply to this

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