### J3(O)o, J3(O)=J4(Q)o, J4(Q)

```F4 + J3(O)o -> E6 + Cx( J3(O)=J4(Q)o ) + U(1) -> E7 + Qx( J4(Q) ) + SU(2) -> E8
|                                                         |
F4         <-          OP2 + B4          <-         OP1 + D4                   ```

B. N. Allison and J. R. Faulkner, in their paper A Cayley-Dickson Process for a Class of Structurable Algebras (Trans. AMS 283 (1984) 185-210), say that 27-dimensional J4(Q)o "has the unique structure of" the 27-dimensional exceptional Jordan algebra J3(O).

As to what "has the unique structure of" means in that context, John Baez and I had an e-mail discussion in November and December 2001, posted on sci.physics.research, that discussed some details:

```John Baez in the thread Re: J3(O) and J4(Q) in sci.physics.research
said:```
```J3(O)o or J4(Q)o ... don't have a multiplicative unit,
which is part of the usual definition of a Jordan algebra.
... in what sense is J3(O) a "subalgebra" of J4(Q)?
```
`I replied: `
```As to exactly in what sense J3(O) is a "sub-thing" of J4(Q),
B. N. Allison and J. R. Faulkner, in their paper
A Cayley-Dickson Process for a Class of Structurable Algebras
(Trans. AMS 283 (1984) 185-210),
state a corollary on page 199:
"Corollary 5.5
Suppose B is
a 28-dimensional central simple Jordan algebra of degree 4
with generic trace t.
Let Bo = { b in B | t(b) = 0 } and
choose e in Bo such that t(e^3) =/= 0.
Then,
Bo has the unique structure of
a 27-dimensional exceptional central simple Jordan algebra
with identity e
and a generic norm given by N_Bo(b) = t(b^3) / t(e^3).".

Since J4(Q) is a 28-dimensional central simple Jordan algebra of degree 4,
and
since J3(O) is a 27-dimensional exceptional central simple Jordan algebra,
I interpret the corollary to imply:

Suppose J4(Q) has generic trace t.
Let J4(Q)o = { b in J4(Q) | t(b) = 0 } and
choose e in J4(Q)o such that t(e^3) =/= 0.
Then,
J4(Q)o has the unique structure of J3(O) with identity e
and a generic norm given by N_J4(Q)o(b) = t(b^3) / t(e^3).

As John Baez said,
Bo (in this case J4(Q)o) is not by itself strictly
speaking a Jordan algebra, but a claim of Allison and Faulkner
(stated in their abstract) is
"... we obtain a procedure for giving the space Bo of trace zero
elements of any such Jordan algebra B the structure of a
27-dimensional Jordan algebra.",
and
they seem to do that (if I understand them correctly, which I recognize
that I may not) by picking out a special element of Bo and
making it act like a unit,
and by modifying the product
(by what they call a Cayley-Dickson process)
so that they end up with a Jordan algebra (in this case J3(O)).

My primary interest in these structures is building a physics model,
and I find in that respect that a chain:

J4(Q) -> J3(O) -> J3(O)o

is useful in that regard (for example, it makes the structure of
the Lie algebras E8, E7, and E6 clearer to me).

I really don't care whether every entry in the chain
J4(Q) -> J3(O) -> J3(O)o
is, according to accepted math terminology, strictly a Jordan algebra,
and
I really don't care whether, in the statement J4(O)o = J3(O),
the = sign means direct isomorphism (which it does not, as you
pointed out), or whether the = sign means something like
"have structures such that one of the structures can
be modified by a natural procedure so that they are isomorphic".

Of course,
my point of view is terrible from the point of view of mathematicians,
because it is based on the subjective term "natural procedure".
However,
it is my opinion that the Allison-Faulkner "procedure" is a
sufficiently natural procedure to allow the chain
J4(Q) -> J3(O) -> J3(O)o
to be used in physics models.

It is a fair criticism of what I said that my terrible mangling
of math terminology is bad in that it corrupts the standards
of math literature (and therefore possibly impedes the clarity
of the subject, making it harder for people to understand it).

On the other hand,
in terms of a physics model it is easier to describe the chain
J4(Q) -> J3(O) -> J3(O)o
as a Jordan algebra chain (while explicitly stating, as I did
in my original message, that I am speaking loosely - the alternative
would be to put in a lot of complicated qualifiers or
to invent new terms such as possibly
Cayley-Dickson-Allison-Faulkner-type Jordan-like algebras)
and
I think that there is a long history of physicists doing such
things to math terminology,
and
a similarly long history of physicists using math things
that are hard to justify rigourously
(look at the difficulties of efforts to construct
"mathematically rigourous" quantum field theories - in their book
PCT, Spin & Statistics, and All That (Benjamin 1964),
Streater and Wightman said "... The physicists who have engaged
in this kind of work are sometimes dubbed the Feldverein.
Cynical observers have compared them to the Shakers,
a religious sect of New England who build solid barns
and led celibate lives, a non-scientific equivalent
of proving rigorous theorems and calculating no cross-sections. ...".).

Anyhow, my main point is that there are interesting and possibly
physically useful relationships among J4(Q), J4(Q)o, J3(O) and J3(O)o,
and I have not seen those things written up anywhere
outside the papers I cited (of Brylinski and Kostant, and
of Allison and Faulkner), and
I would like to see more discussion and analysis of those things.
```
```John Baez said, in a post Re: J3(O) and J4(Q) on sci.physics.research,
about my (very incomplete) description of how J4(Q)o goes to J3(O):```
```this leaves out the really interesting part:
how we define the Jordan algebra structure on these traceless guys
in terms of the Jordan algebra structure on J4(Q)!
```
`I replied:`
```Allison and Faulkner in their paper
A Cayley-Dickson Process for a Class of Structurable Algebras
(Trans. AMS 283 (1984) 185-210)
do describe the "interesting part", and I will try here to
outline what they do (omitting some details to avoid just
reprinting their paper), but please recognize that I may
misunderstand some of their stuff and/or explain it poorly.

That said, here goes:

After proving their Corollary 5.5,
they go at the "interesting part" in Proposition 5.6
(here I will specialize to J3(O) and J4(Q) their more general statements):

Start with J = J3(O) with norm N_J, trace T_J, and identity 1_J.
Let B = F + J (direct sum with field F) as algebras.
Let e = ( 3/2 , -(1/2) 1_J )
Let t be generic trace.
Then
t(e^3) = 3
and
the map j -> j~ = ( (1/2) T_J(j) , j - (1/2) T_J(j) 1_J ) is
a linear bijection on J onto Bo such that N_Bo(j~) = N_J(j) for j in J.

For 27-dim J = J3(O), B = J4(Q) is 1+27=28-dim  and Bo = J4(Q)o is 27-dim,
and
the map j -> j~ of J3(O) onto J4(Q)o is the "interesting" map.

They then give an alternate description of the algebra B = J4(Q)
in this way:

Let X : J3(O) x J3(O) -> J3(O) be defined by

j x k = 2j*k - T_J(j) k - T_J(k) j + ( T_J(j) T_J(k) - T_J(j,k) ) 1_J

where * is the product on J3(O) (in the equation above I wrote
the term T_J(j,k) with a comma as in the paper, but I think that
is probably a misprint and should be T_J(j*k) but maybe I am wrong
about the misprint so I am effectively saying it both ways here
so that you can make up your mind yourself).

They then say that "one easily checks that":

j~ k~ = (1/4) T_J(j,k) 1_B + (1/2) ( j x k )~

[where again I think that T_J(j,k) should really be T_J(j*k)]

for j,k in J3(O) and 1_B = ( 1, 1_J ).

Then,
B = F1_B + Bo (direct sum) (as vector spaces)
1_B is the identity of B = J4(Q)
and
the multiplication on Bo = J4(Q)o = { j~ | j in J } is given by

j~ k~ = (1/4) T_J(j,k) 1_B + (1/2) ( j x k )~

[ which I think should really be

j~ k~ = (1/4) T_J(j*k) 1_B + (1/2) ( j x k )~    ]

Finally, they say e = 1~_J.

-------------------------------------------------------

Note that their construction described above of how J3(O) fits
inside J4(Q) is not obviously (to me at least) a generalized
Cayley-Dickson procedure.

I think that the J3(O) inside J4(Q) stuff is something that
Allison and Faulkner discovered/invented while working on
something that is more clearly a generalized Cayley-Dickson procedure,
that is,
construction of the 56-dimensional irreducible module for E7
from two copies of the 28-dimensional J4(Q) which appears when you
use the Zorn-type construction of the 56-dim E7 module
from 2x2 matrices with two real diagonal entries (field F)
and two off-diagonal 27-dim J3(O) entries.

Allison and Faulkner refer to the analogy with the Zorn construction
of split octonions described on page 142 of Jacobson's book on
Lie algebras.```

### Jordan Algebras and Severi Varieties

"... We study the varieties of reductions associated to the four Severi varieties, the first example of which is the Fano threefold of index 2 and degree 5 studied by Mukai and others. We prove that they are smooth but very special linear sections of Grassmann varieties, and rational Fano manifolds of dimension 3a and index a+1, for a = 1, 2, 4, 8. We study their maximal linear spaces and prove that through the general point pass exactly three of them, a result we relate to Cartan's triality principle. We also prove that they are compactifications of affine spaces. ...

... The most natural possible analogs of the projective space are the Severi varieties AP2 - the Veronese models of the projective planes over the 4 complexified composition algebras A = R,C,H,O, and the varieties of Scorza - the projective n-spaces APn, n > 3 over A = R,C,H ... The ambient projective spaces of all the varieties APn - of Severi, as well of Scorza - are projectivized prehomogeneous spaces. The representation spaces supporting the varieties APn, n = 2, a = 1, 2, 4, 8 and n > 3, a = 1, 2, 4 are exactly the spaces of all the simple complex Jordan algebras P(Jn+1(A)) of rank n > 3. In particular, for A = R = C one obtains again the Jordan algebra J(n+1)(R) = Sym^2 C^(n+1) of symmetric matrices of order n + 1. ...

... For any a = 1, 2, 4, 8, the Jordan algebra J3(A) is the complex vector space of A-Hermitian matrices of order 3. Its projectivization contains three types of matrices, depending on the rank (which can be defined properly even over the octonions). In particular, the (projectivization of the) set of rank one matrices is the Severi variety Xa, the projective A-plane, a homogeneous variety of dimension 2a. For a point w in PJ3(A) defined by a rank three matrix, a non-singular reduction of w is a 3-secant plane to Xa through w ... The projection from the fixed point w sends the quasiprojective set Y_a^0 of nonsingular reductions of w isomorphically to the family of 3-secant lines to the projected Severi variety Xa inside the projective space PJ3(A)o = P(3a+1) of traceless matrices; and the projective closure Y_a of Y_a^0 in the Grassmannian G(2,J3(A)o) = G(2,(3a+2)) is the variety of reductions of w, our main object of study. ...

... we focus on the beautiful geometry of the completion Y_a of Y_a^0, the variety of reductions. This subvariety of G(2,J3(A)0) has dimension 3a, and is endowed with a natural action of the automorphism group SO3(A) := Aut(J3(A)) of the Jordan algebra. We prove that Ya has four SO3(A)-orbits ... This leads to a very nice geometric picture of the Lie algebra isomorphism so3(A) = t(A) + A1 + A2 + A3 ... This geometric occurence of triality completes the picture given by E. Cartan in his paper on isoparametric families of hypersurfaces, the first geometric appearance of the exceptional group F4 ...

... Theorem 3.21. The degrees of the varieties Y_a, and of the Grassmannians G(2,J3(A)0) are:

```deg Y1 = 5                degG(2, 5) = 5
deg Y2 = 57               degG(2, 8) = 132
deg Y4 = 12 273           degG(2, 14) = 208 012
deg Y8 = 1 047 361 761    degG(2, 26) = 1 289 904 147 324 ...".```

"... we give a detailed description of the Chow ring of the complex Cayley plane ... Not to be confused with the real Cayley plane F4/Spin9, the real part of OP2 , which admits a cell decomposition R0 u R8 u R16 and is topologically much simpler. ... X8 = OP2, the fourth Severi variety. This is a smooth complex projective variety of dimension 16, homogeneous under the action of the adjoint group of type E6. It can be described as the closed orbit in the projectivization P26 of the minimal representation of E6 ... For this, we describe explicitely the most interesting of its Schubert varieties and compute their intersection products. Translating our results in the Borel presentation, i.e. in terms of Weyl group invariants, we are able ... to compute the degree of the variety of reductions Y8 in P272 introduced in [math.AG/0306328] ...

... The space J3(O) ... of O-Hermitian matrices of order 3, is the exceptional simple complex Jordan algebra ... The subgroup SL3(O) of GL(J3(O)) consisting in automorphisms preserving the determinant is the adjoint group of type E6. The Jordan algebra J3(O) and its dual are the minimal representations of this group. The action of E6 on the projectivization PJ3(O) has exactly three orbits:

• the complement of the determinantal hypersurface,
• the regular part of this hypersurface, and
• its singular part which is the closed E6-orbit.

These three orbits are the sets of matrices of rank three, two, and one respectively ... The closed orbit, i.e. the (projectivization of) the set of rank one matrices, is the Cayley plane. ... Since the Cayley plane is a closed orbit of E6, it can also be identified with the quotient of E6 by a parabolic subgroup, namely the maximal parabolic subgroup defined by the simple root a6 in the notation below. The semi-simple part of this maximal parabolic is isomorphic to Spin10. ... Schubert cycles in OP2 are indexed by a subset W0 of the Weyl group W of E6, the elements of which are minimal length representatives of the W0- cosets in W. Here W0 denotes theWeyl group of the maximal parabolic P6 in E6: it is the subgroup of W generated by the simple reflections s1, . . . , s5, thus isomorphic to the Weyl group of Spin10. ... The degree of OP2 in P26 is 78. This is precisely the dimension of E6. Is there a natural explanation of this coincidence ? ...

... The Cayley plane OP2 = E6/P6 in PJ3(O) is one of the E6-grassmannians, if we mean by this a quotient of E6 by a maximal parabolic subgroup. ...

... the restriction of J3(O) to the Levi part L =~ Spin(10)xC* of the parabolic subgroup P6 of E6 ... give[s] 1 + 16 + 10 = 27 dimensions, ... the full decomposition [of J3(O)] ...

... In terms of Schubert cycles, the Chern classes of the normal bundle to [complex]OP^2 [in] PJ3(O) are: c_1(N) = 15 H ...

... The degree of the variety of reductions Y8 is degY8 = 1 047 361 761. ...".