The n-Category Café

I don’t know if it’s relevant for you, but the two factors in the SU(5) x SU(5) in E_8 are different, in the sense that the automorphism that switches them doesn’t extend to one of E_8.

As a group, it’s (SU(5) x SU(5))/Z_5, where the Z_5 is generated by (exp(2 pi i/5), exp(4 pi i/5)). Note that this group still has a central Z_5; indeed, it’s the centralizer of that element of E_8 of order 5.

You can see it on the Dynkin diagram level by extending E_8 to the affine diagram, then removing a vertex next to the trivalent, leaving two A_4 diagrams over. That vertex connects to the end of one A_4, and the middle of the other, which leads to the asymmetric amalgamation picture above.

The embedding of $a_4\times a_4 \subset e_8$ was the basis of my (embarrassing) failed attempt to find 2 generations in your model.

The more general argument, that it’s impossible to get even 2 generations is independent of any of the details of how the Standard Model is embedded in $E_8$. It relies on Berger’s classification of involutions of $e_8$ and on the fact that the center of $SL(2,\mathbb{C})$ acts as $-1$ on fermions and as $+1$ on bosons (so that any embedding of $SL(2,\mathbb{C})\subset E_8$ provides such an involution).

Yeps, very nice talk. I didn’t know Lie algebras were related to densely packing spheres! I am also ashamed to admit I learnt about roots and weights the “old-school Lie-algebra-only” way; no-one told me/stressed that the lattices $A_n$, $D_n$, $E_8$ and so on are just the kernel of the exponential map on the maximal torus, which is definitely helpful.

I’m glad you liked my talk.

No-one told me/stressed that the lattices $A_n$, $D_n$, $E_8$ and so on are just the kernel of the exponential map on the maximal torus, which is definitely helpful.

Thanks! If you like this approach to classifying simple Lie algebras — thinking a lot about lattices, and bringing Lie groups into the game early instead of avoiding them out of some misguided fanatical devotion to ‘purely algebraic’ methods — I strongly recommend Frank Adams’ book Lectures on Lie Groups. It was only after I learned this approach that I grew to love simple Lie algebras.

Or, if you’re in a rush, try week63 — week65, which I wrote shortly after I overcame my early hatred of roots and weights.

You can think of my talk as a miniaturized version of those Weeks, with a ridiculous overemphasis on $E_8$

Haven’t had time to view the lecture
Thanks for the heads up
Can you clue me in as to page or how far in I’ll find half my beloved

Daniel wrote:

So, Konstant’s result was the representation 5 ⊗ 10 ⊕ 5* ⊗ 10* ⊕ 10 ⊗ 5 ⊕ 10* ⊗ 5*, right?

I don’t know if that’s new. Experts like Allen Knutson already knew — or could quickly see — that $su(5) \times su(5)$ sits inside $e_8$. As a trained professional, once you know something like this, it’s straightforward to work out the representation of $su(5) \times su(5)$ that this embedding gives.

So, while that result is pretty, I suspect the most novel part of Kostant’s talk is what he spent the most time talking about: the way various interesting groups arise inside $E_8$ as centralizers of finite subgroups, and also how characters of the $(\mathbb{Z}/2)^5$ subgroup allow you to chop $e_8$ into 31 ‘pieces of eight’.

(Yes, I know I’m abusing the original meaning of ‘pieces of eight’.)

Don’t forget the letter written by E8 himself to counter the canard that he is too hard to understand and one should work up through E6 and E7.

Lutz wrote:

Unfortunately, especially for newcomers, the Lie groups/ Lie algebra textbook world is such a disaster. Not only do most of the 1001 Lie groups books out there prefer talking about Haar measure than about Dynkin diagrams and exceptional groups. But also the weights and roots material is often obscured behind all kinds of abstract language.

I know what you mean. I’ve actually loved Haar measure and other general concepts in Lie theory since I was a student, but it took me quite a while to penetrate the veil and learn to love the semisimple theory — roots, weights, Cartans, Borels, and all that jazz. Why? At least for me, these turned out to be more beautiful in concrete examples than in abstract generality… but my first introduction to them never properly explained the examples!

What could be cooler than a way of packing spheres in 4 dimensions built by first packing spheres in a cubical lattice as tightly as possible, and then discovering there’s still enough room to slip another copy of this cubical lattice of spheres in the cracks between the first one? And then to discover that when you’ve done this, each sphere touches 24 others: 16 sitting at the vertices of a hypercube, and 8 sitting at the vertices of a hyperoctahedron? And then, to realize that these 24 vertices are the corners of a 4-dimensional Platonic solid, the 24-cell — the sixth and final one, the one that makes there be more Platonic solids in 4 dimensions than any other dimension? And then, to realize that the lattice you’ve got can be thought of as consisting of integer quaternions? And then, to discover that this lattice is the root lattice of the group $Spin(8)$, the double cover of the rotation group in 8 dimensions? And then, to discover that the symmetry of permuting the quaternions $i, j,$ and $k$ gives this lattice and thus $Spin(8)$ a special symmetry called ‘triality’, which no other Spin group has? And then, to discover that triality gives rise to the octonions? And then … well, the wonders never cease!

So, you have to penetrate the veil and see some of this beauty firsthand!

Someday I want to write a book called Exceptional Beauty, which will be a guided tour of all this material.

But for now, the main books I recommend are Adams’ Lectures on Lie Groups and Fulton and Harris’ Representation Theory — A First Course. Start with the pictures and work your way out.

Also, read all of This Week’s Finds!

That paper by Adler looks fun, and I urge people interested in grand unified theories to take a peek. As some of you know, this is the Stephen Adler who gained fame for the Adler–Bell–Jackiw anomaly, got a job at the Institute for Advanced Studies, and then spent many years working on quaternionic quantum mechanics — a subject most physicists regard as a lost cause.

(I’ve spent plenty of time on QQM myself, so this isn’t a criticism of Adler: I’m just reporting a sociological fact! But, I part company with Adler when he starts tensoring ‘quaternionic vector spaces’ that are defined to be left modules of the quaternions — I think one really should define them to be bimodules, since the quaternions are a noncommutative ring! My former student Toby Bartels succesfully worked out a fair amount of quaternionic functional analysis after realizing this fact.)

Anyway, the first thing to note is that Adler is studying supersymmetric $E_8$ Yang–Mills theory, so he’s not trying to pack both fermions and bosons into $E_8$ the way Lisi is — and he’s not trying to include gravity, either.

He’s not trying to pack both fermions and bosons into $E_8$… or is he? I’m a bit confused: he says “in this $E_8$ model the fermions and gluons [gauge bosons] are in the same supermultiplet, achieving a complete unification of matter fields and force-carrying fields. The point that an $E_8$ unification is automatically supersymmetric was made independently more than twenty years by… [lots of guys].”

Anyway, apart from this remark, he seems to be considering fermions and bosons separately, but both in the adjoint representation. He uses an embedding

$$su(3) \times \mathrm{e}_6 \subset \mathrm{e}_8$$

together with the usual embedding

$$so(10) \times \mathrm{u}(1) \subset \mathrm{e}_6$$

to fit the 16-dimensional spinor rep of $so(10)$ (familiar from the $so(10)$ GUT) into the 27-dimensional rep of $\mathrm{e}_6$ (which means lots of extra unseen fermions) and then get three generations of these fermions related by $su(3)$ symmetry.

However, he writes, “Despite the attractive features of automatic supersymmetry and natural inclusion of three families, the reason that $E_8$ has not been further pursued as a unification group is that in addition to three families, it also contains three mirror families. Thus, under $E_8 \supset SU(3) \times E_6$, in addition to three $27$’s there are three $\overline{27}$’s. The presence of mirror families leads to potential phenomenological and theoretical difficulties.”

Another depressing but cute remark is this: he says an asymptotically free $E_8$ gauge theory can’t do symmetry-breaking via a Higgs, because the Dynkin index of the smallest candidate Higgs is already too large — it’s a whopping 3875-dimensional rep.

Okay — so the asymmetry between the two $SU(5)$’s is manifest at the Lie algebra level. I just watched Kostant’s video again, and this part is right near the end. It’s sort of amusing: he says $\overline{10} \otimes 5$, but he writes $\overline{10} \otimes \overline{5}$ — and that’s what I copied down. It seemed nice and symmetrical!

I wish I were quicker at calculations like this: I would like to see for myself how it works.

The people at the media center, who run the computers where the downloadable videos are stored, accidentally made it impossible to access the videos without a password. I told them about this and they are trying to fix it.

When I last checked, the streaming videos were working… but perhaps they are not so convenient for you.

I had problems with streaming and download too. If someone managed to download ok, maybe they can upload it to YouTube

Thank you! Download is working. Streaming still makes my browser crash, but that’s ok, since I can dowload the file.

Best wishes,
Christine

rntsai mentioned Kostant’s “… decomposition of e8 into 31 cartan …” and said:
“… related to e8/(d4+d4) decomposition :
e8/(d4+d4)= (28,1)+(1,28) + (8v,8v) + (8S+,8S+) + (8S-,8S-)
The last 3 terms can be seen as 24 8-dim spaces invariant under one of the d …
The other 7 cartans are inside d4+d4 … there are probably several ways to identify [them]…”.

Another way (other than the one mentioned by rntsai) is to
decompose d4 into a 14-dim G2 plus two 7 spheres S7 + S7, getting
d4 = 14 + 7 + 7

14-dim rank-2 G2 has 7 = 14/2 Cartans
and G2 can be seen as the sum of two 7-dimensional representations.
If each 7 is represented by the 7 imaginary octonions { i,j,k,e,ie,je,ke }
then the 7 Cartans of G2 are the 7 pairs (one from each of the 7 in G2):
i i
j j
k k
e e
ie ie
je je
ke ke

Note that to make an Abelian Cartan, the pairs must match, because only matching pairs close to form an Abelian Cartan (this can be seen by looking at the octonion products).

28-dim rank-4 d4 has 28/4 = 7 Cartans
and d4 looks like G2 plus S7 plus S7
and since G2 decomposes into two 7 representations
d4 decomposes into 7 + 7 + 7 + 7 ( where the first two 7 are from G2 and the other two come from the two S7 )
and
the 7 Cartans of d4 are ( in terms of octonion imaginaries )
i i i i
j j j j
k k k k
e e e e
ie ie ie ie
je je je je
ke ke ke ke

Again, note that all elements of the quadruples must match to get Abelian Cartan structure.

When you look at d4 + d4 to get 8-element Cartans of E8, all 8 elements must again match up to get Abelian Cartan structure, so the 8 Cartans of E8 that come from d4 + d4 look like
i i i i i i i i
j j j j j j j j
k k k k k k k k
e e e e e e e e
ie ie ie ie ie ie ie ie
je je je je je je je je
ke ke ke ke ke ke ke ke

Of course, this octonion structure is also reflected in the
“… 24 8-dim spaces …” described by rntsai as
“… (8v,8v) + (8S+,8S+) + (8S-,8S-) …”

so that all 31 of the 8-dim Cartans of E8 have nice octonionic structure.

Also note that when you make a 240-element E8 root vector diagram of 8 circles each with 30 vertices, 8 of the 248 E8 generators are missing, so that you must leave out one of the 31 Cartan 8-element sets.
Seeing E8 in terms of E8 = 120 + 128 = d4 + d4 + 8x8 + 8x8 + 8x8
it is most natural to see the Cartan as being one of the Cartan sets of 8 coming from the d4 + d4,
but you could see the E8 from other points of view by using other Cartan sets of 8 to determine which of the 248 were the 8 omitted from the root vector diagram.

Tony Smith

“When you look at d4 + d4 to get 8-element Cartans of E8, all 8 elements must again match up to get Abelian Cartan structure, so the 8 Cartans of E8 that come from d4 + d4 look like
i i i i i i i i
j j j j j j j j
…”

I’m afraid I don’t see how this construction with octanions is done.
Are you building e₈ with 8-tuples of octanions? so iiiiiiii is
{(i,0,…,0),(0,i,…,0),…(0,…,i)}?
what happens with the other combinations?
there are too many to fit into the 248
roots of e₈

Uhh, allow me to correct/expand my comment. When you add colour is not the same that when you add U(3). So that 54=15+15+24 will, when U(3) hits its fan, smells as 54*9 = 486. Two singlets seem lost in translation, and it could be $$496-1 = (54+1) \times 9$$

So, the $SU(5)$ of flavour in the model of hadrons uplifts -bootstraps, say- itself up to an object of 495 elements, but I am not sure if it is an E8 or what. Under $U(3)$ it should break as:

$$496 - 1 = (24,1) + (24,8) +(1,1)+ (1,8) + (15,3)+(15,6) + (\bar {15}, \bar 3) + (\bar {15}, \bar 6)$$

where $(24,1), (15,3) ,(\bar {15}, \bar 3)$ are the plets discussed in the previous (9 years ago!) comment.