` `

# ZeroDivisor Tensor Algebras:

| CxS = 32 | QxS = 64 | ImQxO = 24 | OxS = 128 | SxS = 256 | del Pezzo Surfaces |

```
Charles Muses has described ZeroDivisor Algebras
in terms of subalgebras of 16-dimensional Sedenions
and their Complexifications.

The Sedenions and their subalgebras are:
(here  a,b  are nonzero)

R  - dimension 2^0 =   1   - real numbers,    with a^2 =  1;
C  - dimension 2^1 =   2   - complex numbers, with a^2 = -1;
Q  - dimension 2^2 =   4   - quaternions;
O  - dimension 2^3 =   8   - octonions.
S  - dimension 2^4 =  16   - sedenions S      with ab  =  0;

The Complexifications of those Algbras are:

CxR  - dimension 2x1  =  2
CxC  - dimension 2x2  =  4
CxQ  - dimension 2x4  =  8
CxO  - dimension 2x8  = 16
CxS  - dimension 2x16 = 32

The Algebras of dimensions 64, 128, and 256
are then constructed by using only Real Matix Algebras:

M(8,R)        - dimension  64
M(8,R)+M(8,R) - dimension 128
M(16,R)       - dimension 256

```

## Another Way to get to Dimensions 64, 128, 256,

```is to follow the approach of Boris Rosenfeld
and consider not only complexifications of algebras
(which Rosenfeld calls bialgebras, such as bioctonions CxO)

but also
tensor products by Quaternions QxA to get quateralgebras
and by Octonions OxA to get octoalgebras.

You can also
do tensor products by Sedenions SxA to get sedenalgebras
if you are willing to put up with the complication
of nonzero a,b such that ab = 0.

One notational change from Muses conventions
that is needed in this approach is to change

1  0                                                  i  0
0  1  = e0                                      i0 =  0  i

to

1  0                                                  i  0
0  1  = i0                                      e0 =  0  i

Then construct up to the Complexified Sedenions as before,
to get

1  0                                                  i  0
0  1  = i0                                      e0 =  0  i

0 -1                   0  i
i1  =  1  0                  -i  0           = e1

i  0                   1  0
i2  =  0 -i                   0 -1           = e2

0  i                   0  1
i3  =  i  0                   1  0           = e3

1  0  |  0 -1          1  0  |  0  i
i4  =  0  1  |  1  0          0  1  | -i  0  = e4

0 -1  |  0 -1          0  1  |  0  i
i5  =  1  0  |  1  0         -1  0  | -i  0  = e5

-i  0  |  0 -1         -i  0  |  0  i
i6  =  0  i  |  1  0          0  i  | -i  0  = e6

0  i  |  0 -1          0  i  |  0  i
i7  =  i  0  |  1  0          i  0  | -i  0  = e7

1  0  |  i  0          1  0  |  1  0
i8  =  0  1  |  0 -i          0  1  |  0 -1  = e8

0 -1  |  i  0          0  1  |  1  0
i9  =  1  0  |  0 -i         -1  0  |  0 -1  = e9

-i  0  |  i  0         -i  0  |  1  0
i10 =  0  i  |  0 -i          0  i  |  0 -1  = e10

0  i  |  i  0          0  i  |  1  0
i11 =  i  0  |  0 -i          i  0  |  0 -1  = e11

1  0  |  0  i          1  0  |  0  1
i12 =  0  1  |  i  0          0  1  |  1  0  = e12

0 -1  |  0  i          0  1  |  0  1
i13 =  1  0  |  i  0         -1  0  |  1  0  = e13

-i  0  |  0  i         -i  0  |  0  1
i14 =  0  i  |  i  0          0  i  |  1  0  = e14

0  i  |  0  i          0  i  |  0  1
i15 =  i  0  |  i  0          i  0  |  1  0  = e15

and

then consider ```

## the 32-dimensional algebra

```to be made of
the i-numbers corresponding to the real axis
and
the e-numbers corresponding to
the Complex i imaginary axis (the tensor products by C):

1    i

i0   e0
i1   e1
i2   e2
i3   e3
i4   e4
i5   e5
i6   e6
i7   e7
i8   e8
i9   e9
i10  e10
i11  e11
i12  e12
i13  e13
i14  e14
i15  e15

Then consider ```

## the 64-dimensional algebra

```to be made of
the i-numbers corresponding to the real axis
and
the e-numbers corresponding to
the Quaternion i imaginary axis (the tensor products by Q)
and
the p-numbers corresponding to
the Quaternion j imaginary axis (the tensor products by Q)
and
the q-numbers corresponding to
the Quaternion k imaginary axis (the tensor products by Q):

1    i    j    k

i0   e0   p0   q0
i1   e1   p1   q1
i2   e2   p2   q2
i3   e3   p3   q3
i4   e4   p4   q4
i5   e5   p5   q5
i6   e6   p6   q6
i7   e7   p7   q7
i8   e8   p8   q8
i9   e9   p9   q9
i10  e10  p10  q10
i11  e11  p11  q11
i12  e12  p12  q12
i13  e13  p13  q13
i14  e14  p14  q14
i15  e15  p15  q15

This may correspond to the introduction of the
p-number and q-nbumber imaginary types of Charles Muses.

Also, it gives another way to look at the 24-dimensional algebra,
as the Quaternion pure imaginary part of QxO,
or the tensor product of the 3-sphere S3 with the Octonions:

i    j    k

e0   p0   q0
e1   p1   q1
e2   p2   q2
e3   p3   q3
e4   p4   q4
e5   p5   q5
e6   p6   q6
e7   p7   q7

Then consider ```

## the 128-dimensional algebra

```to be made of
the i-numbers corresponding to the real axis
and
the e-numbers corresponding to
the Octonion i imaginary axis (the tensor products by O)
and
the p-numbers corresponding to
the Octonion j imaginary axis (the tensor products by O)
and
the q-numbers corresponding to
the Octonion k imaginary axis (the tensor products by O)
and
the w-numbers corresponding to
the Octonion E (tensor product by O)
and
the Omega(W)-numbers corresponding to
the Octonion I imaginary axis (the tensor products by O)
and
the m-numbers corresponding to
the Octonion J imaginary axis (the tensor products by O)
and
the v-numbers corresponding to
the Octonion K imaginary axis (the tensor products by O):

1    i    j    k    E    I    J    K

i0   e0   p0   q0   w0   W0   m0   v0
i1   e1   p1   q1   w1   W1   m1   v1
i2   e2   p2   q2   w2   W2   m2   v2
i3   e3   p3   q3   w3   W3   m3   v3
i4   e4   p4   q4   w4   W4   m4   v4
i5   e5   p5   q5   w5   W5   m5   v5
i6   e6   p6   q6   w6   W6   m6   v6
i7   e7   p7   q7   w7   W7   m7   v7
i8   e8   p8   q8   w8   W8   m8   v8
i9   e9   p9   q9   w9   W9   m9   v9
i10  e10  p10  q10  w10  W10  m10  v10
i11  e11  p11  q11  w11  W11  m11  v11
i12  e12  p12  q12  w12  W12  m12  v12
i13  e13  p13  q13  w13  W13  m13  v13
i14  e14  p14  q14  w14  W14  m14  v14
i15  e15  p15  q15  w15  W15  m15  v15

This may correspond to the introduction of the
w-number, Omega(W)-number, m-number, and v-number
imaginary types of Charles Muses.

Then consider ```

## the 256-dimensional algebra

```to be made of
the i-numbers corresponding to the real axis
and
the e-numbers corresponding to
the Sedenion i imaginary axis (the tensor products by S)
and
the p-numbers corresponding to
the Sedenion j imaginary axis (the tensor products by S)
and
the q-numbers corresponding to
the Sedenion k imaginary axis (the tensor products by S)
and
the w-numbers corresponding to
the Sedenion E (tensor product by S)
and
the Omega(W)-numbers corresponding to
the Sedenion I imaginary axis (the tensor products by S)
and
the m-numbers corresponding to
the Sedenion J imaginary axis (the tensor products by S)
and
the v-numbers corresponding to
the Sedenion K imaginary axis (the tensor products by S):
and
8 more types of imaginary numbers corresponding to
the Sedenion S,T,U,V,W,X,Y,Z imaginary axes
(the tensor products by S)
but here, denote the S,T,U,V,W,X,Y,Z imaginary axes by
1'   i'   j'   k'   E'   I'   J'   K'
because they correspond to the axes
1    i    j    k    E    I    J    K
of the octonion imaginaries plus real axis:

1    i    j    k    E    I    J    K    1'   i'   j'   k'   E'   I'   J'   K'

i0   e0   p0   q0   w0   W0   m0   v0   i'0  e'0  p'0  q'0  w'0  W'0  m'0  v'0
i1   e1   p1   q1   w1   W1   m1   v1   i'1  e'1  p'1  q'1  w'1  W'1  m'1  v'1
i2   e2   p2   q2   w2   W2   m2   v2   i'2  e'2  p'2  q'2  w'2  W'2  m'2  v'2
i3   e3   p3   q3   w3   W3   m3   v3   i'3  e'3  p'3  q'3  w'3  W'3  m'3  v'3
i4   e4   p4   q4   w4   W4   m4   v4   i'4  e'4  p'4  q'4  w'4  W'4  m'4  v'4
i5   e5   p5   q5   w5   W5   m5   v5   i'5  e'5  p'5  q'5  w'5  W'5  m'5  v'5
i6   e6   p6   q6   w6   W6   m6   v6   i'6  e'6  p'6  q'6  w'6  W'6  m'6  v'6
i7   e7   p7   q7   w7   W7   m7   v7   i'7  e'7  p'7  q'7  w'7  W'7  m'7  v'7
i8   e8   p8   q8   w8   W8   m8   v8   i'8  e'8  p'8  q'8  w'8  W'8  m'8  v'8
i9   e9   p9   q9   w9   W9   m9   v9   i'9  e'9  p'9  q'9  w'9  W'9  m'9  v'9
i10  e10  p10  q10  w10  W10  m10  v10  i'10 e'10 p'10 q'10 w'10 W'10 m'10 v'10
i11  e11  p11  q11  w11  W11  m11  v11  i'11 e'11 p'11 q'11 w'11 W'11 m'11 v'11
i12  e12  p12  q12  w12  W12  m12  v12  i'12 e'12 p'12 q'12 w'12 W'12 m'12 v'12
i13  e13  p13  q13  w13  W13  m13  v13  i'13 e'13 p'13 q'13 w'13 W'13 m'13 v'13
i14  e14  p14  q14  w14  W14  m14  v14  i'14 e'14 p'14 q'14 w'14 W'14 m'14 v'14
i15  e15  p15  q15  w15  W15  m15  v15  i'15 e'15 p'15 q'15 w'15 W'15 m'15 v'15

This 256-dimensional algebra corresponds to M(16,R),
the Clifford Algebra Cl(8) of 16x16 real matrices,
whose 16-dimensional full spinor space is reducible
to two 8-dimensional half-spinor spaces.

The full 16-dimensional row spinor space is represented by

1    i    j    k    E    I    J    K    1'   i'   j'   k'   E'   I'   J'   K'
or
i0   e0   p0   q0   w0   W0   m0   v0   i'0  e'0  p'0  q'0  w'0  W'0  m'0  v'0

which reduces to the two 8-dimensional half-spinor spaces

1    i    j    k    E    I    J    K
or
i0   e0   p0   q0   w0   W0   m0   v0

and

1'   i'   j'   k'   E'   I'   J'   K'
or
i'0  e'0  p'0  q'0  w'0  W'0  m'0  v'0

Since the two 8-dimensional half-spinor spaces are isomorphic
to each other (as well as the Cl(8) vector space, by triality),
no qualitatively new imaginary types are introduced,
and, as Charles Muses says, there are only 8 types of imaginaries:

i    e    p    q    w    W    m    v

corresponding to the 8 half-spinors

1'   i'   j'   k'   E'   I'   J'   K'
or
i0   e0   p0   q0   w0   W0   m0   v0

In the D4-D5-E6-E7 physics model
the space-time gammas
are represented by the row spinor space
while
the 8 first-generation fermion particles
and
the 8 first-generation fermion antiparticles
are represented by the column spinor space.

By isomorphism, you could equivalently
represent the 8 first-generation fermion particles
and
the 8 first-generation fermion antiparticles
by the row spinor space

1    i    j    k    E    I    J    K    1'   i'   j'   k'   E'   I'   J'   K'

and represent the space-time gammas
by the column spinor space

i0
i1
i2
i3
i4
i5
i6
i7
i8
i9
i10
i11
i12
i13
i14
i15

Charles Muses relates the 8 dimensions of Octonions to ```

## del Pezzo Surfaces and the A-D-E series.

What would Octonionic del Pezzo Surfaces look like?
```In Applied Mathematics and Computation 60:25-36 (1994),
Charles Muses says (at page 28):
"... In the late 19th century, it was the Neapolitan mathematician
P. del Pezzo who first introduced the brilliant idea of a del Pezzo
(hyper)surface: one described by a function of nth degree in n-space.
Thus, a del Pezzo surface in 5-space could explain the 27 straight
lines on the general cubic surface. Del Pezzo's discovery also
hinted at the special Lie algebras and hence at octonions since they
climaxed in an 8th degree variety in 8-space, linking with the fact
that it is E8, the highest special Lie algebra, that is intimately
connected with octonions. Indeed, the whole theory of Lie algebras
depends ultimately on its elemental core of octonion arithmetic.
... The entire Italian school at that period was exploring higher
dimensional geometry ...".

A del Pezzo Surface F(N,2) is a surface of order N
in N-dimensional real space.

del Pezzo noted that any algebraic surface must be
either a cone or other ruled surface
or a non-ruled surface

and that

F(3,2), the cubic surface in 3-space,
has 27 lines, each of which is skew to 16 others,
and maximal sets of 6 skew lines.  ``` ```and that

projecting from F(4,2) to F(3,2) would project
to 6 skew lines from 5 skew lines, so that the
maximal sets of skew lines in F(4,2) have 5 skew lines.

Similarly,
maximal sets of skew lines in F(5,2) have 4 skew lines;
maximal sets of skew lines in F(6,2) have 3 skew lines;
maximal sets of skew lines in F(7,2) have 2 skew lines;
maximal sets of skew lines in F(8,2) have at most 1 skew line;
and
there are no skew lines in F(9,2).

Since a non-ruled surface F(10,2) would yield,
by projection, a line on F(9,2),
NON-RULED SURFACES ONLY OCCUR WHEN N IS NO GREATER THAN 9.

Here is Coxeter's description of a correspondence
between the lines of del Pezzo surfaces
and polytopes and symmetry groups of those polytopes:

N  Lines            Polytope         Dim  Symmetry Group

9    0                 0              0     A0 U(1)

8    1 or 0       line segment        1     A1 SU(2)

7    3              triangle          2     A2 SU(3)

6    6           triangular prism     3     A2xA1 SU(3)xSU(2)

5   10      4simplex edge-midpoints   4     A4 SU(5)

4   16             Gosset 1_21        5     D5 Spin(10)

3   27             Gosset 2_21        6     E6

2   56=28+28       Gosset 3_21        7     E7

1  240        Witting = Gosset 4_21   8     E8

0  infinite        Gosset 5_21        9     E9 = affine
extension of E8

I prefer different correspondences at the levels N = 6 and 5,
producing the following table in which the isomorphism A3=D3
provides a natural transition from the A series to the D series
in the A-D-E series:

N  Lines            Polytope         Dim  Symmetry Group

9    0                 0              0     A0 U(1)

8    1 or 0       line segment        1     A1 SU(2)

7    3              triangle          2     A2 SU(3)

6    6    cuboctahedron diagonals     3     A3=D3 SU(4)=Spin(6)

5   10  6+4 special lines of 24-cell  4     D4 Spin(8)

4   16     Gosset 1_21 (half-5cube)   5     D5 Spin(10)

3   27             Gosset 2_21        6     E6

2   56=28+28       Gosset 3_21        7     E7

1  240        Witting = Gosset 4_21   8     E8

0  infinite        Gosset 5_21        9     E9 = affine
extension of E8

In my preferred correspondences, the 6 coboctahedron diagonals are and there are 6+4 = 10 special lines of the 24-cell: the 6 diagonals of a central cuboctahedron and3+1 lines of the 2 (red and blue) octahedra which, with the (green) central cuboctahedron, make the 24-cell: the 3 red octahedra diagonals and the 3 blue octahedra diagonals are parallell and so are identified as 3 special lines and the cyan line of offset of the red and blue octahedra is identified as a 4th special line which when added to the 6 diagonal lines of the central cuboctahedron give the 3+1+6 = 4+6 = 10 special lines of a 24-cell
To see how this is geometrically similar
to Coxeter's 10 4-simplex edge-midpoints, visualize ``` ```the 4simplex as 6 blue edges of a tetrahedron
plus 4 red edges meeting at a central vertex,
and the 6 blue edge-midpoints corresponding to the 6 cuboctahedron diagonalsand 3 of the red edge-midpoints corresponding to the 3 parallel octachedral diagonals and the 4th red edge-midpoint corresponding to the offset of the two 24-cell octahedra.

Whichever correspondences you prefer,
the following descriptions are applicable:

For N = 4, the Gosset polytope 1_21 is a half-5cube.

For N = 3, the Gosset polytope 2_21 has the same
symmetry as the Weyl group of the E6 Lie Algebra,
as is described in
the Gem of the Modular Universe by Bruce Hunt.

For N = 2, the 28 diameters of the Gosset polytope 3_21
correspond to the 28 bitangents of the quartic curve
arising from projecting the del Pezzo cubic surface onto a plane,
and to the 28-dimensional Lie algebra Spin(8).

For N = 1, the 120 diameters of the Witting polytope
correspond to the 120 tritangent planes of the sextic space
of the intersection of a cubic surface with two sheets
of a conical surface.

For N = 0, the Gossett 5_21 is not a finite polytope,
but is the E8 lattice, which is made up of repeated
configurations of two types of 8-dimensional polytopes:
128 simplexes and 9 cross-polytope hyperoctahedra,
for 137 polytopes per configuration.

Since each of 27 lines of F(3,2), the cubic surface in 3-space,
is skew to 16 others, there are 16 lines on F(4,2).
Also:
each line on F(4,2) has 10 skew lines, so F(5,2) has 10 lines;
each line on F(5,2) has  6 skew lines, so F(6,2) has  6 lines;
each line on F(6,2) has  3 skew lines, so F(7,2) has  3 lines.
F(7,2) has two kinds of skew sets (1 or 0),
so there are two kinds of F(8,2) with 0 lines
or with 1 line having 2 orientations.

Coxeter notes that
the product  240 x 56 x 27 x 16 x 10 x 6 x 3 x 1
of the number of lines from N = 1 to N = 8
is related to
the product  240 x 56 x 27 x 16 x 10 x 6 x 2 x 1
of the order of the Weyl Group of the Lie Algebra E8.
Perhaps
the factor 2 in the Weyl Group of E8
is due to the fact that the triangle for N = 7
(from which the factor 3 comes) is an isosceles triangle,
rather than an equilateral triangle.  ```

Robert Friedman and John W. Morgan in math.AG/0009155 describe Exceptional Groups and del Pezzo Surfaces.

### What would Octonionic del Pezzo Surfaces look like?

"... Does anyone know whether there is a natural definition of an "octonionic blow-up" which could be used to construct an octonionic del Pezzo surface?  ...

... The definition of a "del Pezzo surface" is that it's a compact two-dimensional nonsingular variety over C with ample anticanonical divisor (morally, "positive curvature.") If we wrote "one-dimensional" instead of "two-dimensional" then that definition would single out P^1 as the unique such object; in two dimensions there are 10 such surfaces instead of 1. So what are these 10 special surfaces? You get nine of them by starting with P^2 and blowing-up k<=8 points in general position (call the surface obtained in this way B_k), and the tenth one is P^1 x P^1. (It turns out that blowing up one point on P^1 x P^1 gives you B_2, not something new.) Anyway, the surface B_k has a lot to do with the exceptional group E_k (suitably defined for k<6). Most of the connections have to do with the fact that blowing up a smooth point produces a curve of self-intersection -1, so that the second homology (with its usual intersection product) is a rectangular lattice of signature (1,k), and furthermore the orthocomplement of the canonical divisor turns out to be exactly the root lattice of E_k. Furthermore the Weyl group of E_k acts by diffeomorphisms. So all kinds of geometrical objects on B_k (lines, rulings, curves of degree 4, whatever) fall into representations of E_k, and there are natural algebraic E_k bundles over B_k, and probably a lot more connections I don't know about.

OK, so why did I want to know about the octonionic version? It's a crazy idea, but if you're curious, here it is. There is a proposed "duality" (hep-th/0111068) which relates some of the objects which appear in M-theory (compactified on a k-dimensional torus) with curves lying on the del Pezzo B_k. In particular, the exalted "M-theory in 11 flat dimensions" gets identified with CP^2 in this duality. The outlier P^1 x P^1 gets identified with the Type IIB string, which is also an outlier from the M-theory point of view. Various numerological things work out well in this duality, and there are enough coincidences to make you feel that it should mean *something*, but only some aspects of M-theory are included so far -- some of the moduli are frozen to zero, and the Lorentz symmetry is hard to see -- so it seems like to get the full story the del Pezzo surface needs to be augmented in some way.

Switching to an octonionic surface seems like overkill at first, since then (assuming everything works out like the complex case) you would roughly be getting the exceptional symmetries *twice* -- once from the octonions and once from the blow-ups -- but this could actually be right, because also in M-theory one seems to find exceptional symmetries occurring in two different ways: there is an E_k which appears on compactification as the U-duality group, and there is another E_8 which appears already in 11 dimensions and is responsible e.g. for the E_8 x E_8 heterotic string. (It could be that these are both somehow the same, but so far nobody has any evidence for that, and actually there is an argument that they can't be the same because the E_k appears in its non-compact form while the E_8 appears in its compact form.) Another reason to think about the octonionic P^2 is that I'm attracted to the idea that the embedding of Spin(9) in F_4 is important for M-theory, because Ramond has discovered a way to get the Lorentz content of M-theory naturally from this embedding (hep-th/0112261). And anyway, these exceptional structures should be good for *something*, right? ...".

In trying to understand Andy Neitzke's interesting ideas, I posted:

"... As Amer Iqbal, Andy Neitzke, and Cumrun Vafa said in http://xxx.lanl.gov/abs/hep-ph/0111068
• "... M-theory on Tk corresponds to P2 blown up at k generic points;
• Type IIB corresponds to P1 x P1.
• The moduli of compactifications of M-theory on rectangular tori are mapped to Kaehler moduli of del Pezzo surfaces.
• The U-duality group of M-theory corresponds to a group of classical symmetries of the del Pezzo represented by global diffeomorphisms.
• The 1/2-BPS brane charges of M-theory correspond to spheres in the del Pezzo, and
• their tension to the exponentiated volume of the corresponding spheres.
• The electric/magnetic pairing of branes is determined by the condition that the union of the corresponding spheres represent the anticanonical class of the del Pezzo.
• The condition that a pair of 1/2-BPS states form a bound state is mapped to a condition on the intersection of the corresponding spheres. ...".

In other words, desingularization/blowup structures on the CP^2 complex projective plane have interesting connections with string theory structures and maybe similar structures on an octonionic projective plane might have more interesting connections with string theory structures.

A question is:

Which of the following do you want to be your generalized octonionic projective plane:

• (RxO)P^2 = OP^2 = F4 / Spin(9) (16-dim "real" octonionic projective plane)
• (CxO)P^2 = E6 / Spin(10)xU(1) (its 32-dim "complexification")
• (HxO)P^2 = E7 / Spin(12)xSp(1) (its 64-dim "quaternification")
• (OxO)P^2 = E8 / Spin(16) (its 128-dim "octonification") ????

(For descriptions of those manifolds, see, for example, Rosenfeld's book Geometry of Lie Groups.)

Being naive about all this stuff, I guess that you might try to generalize what is said on page 321 of Einstein Manifolds by (pseudonymous) Arthur L. Besse:

"... If M is a complex surface (m=2) ...[and]... c_1(M) is positive, negative or zero, the Chern number c_1^2(M) is non-negative ... blowing up a point ... ... by replacing a point p of M by the set of (complex) tangent directions around the point, leaving unchanged the remainder of M, ...[gives]... a new complex manifold M~ and a holomorphic mapping from M~ to M, bilholomorphic over M - {p}, and having a fiber isomorphic to CP^(m-1) over p ... the fiber over p is called the exceptional divisor of the blowing up. ... the blowing up process decreases the Chern number c_1^2. In particular, starting from the complex projective plane CP^2 ... for which c_1^2 is equal to 9 ... by blowing up 9 points or more we certainly obtain a complex surface whose first Chern class is indefinite. ... the complex surfaces S_r obtained from CP^2 by blowing up r distinct points, 0 <= r <= 8, do have a positive first Chern class, whenever those points are in general position ... ...[they]... are the only (compact) complex surfaces having positive first Chern class, with CP^2 and CP^1 x CP^1 ...".

Another question is:

Is the property of holomorphy of the mapping M~ to M important for your generalization, and if so, then how do you define holomorphy for the octonionic structures?

(For example, would you generalize the Cauchy-Riemann equations in the manner of section 4.2 of Nonassociative Algebras in Physics by Lohmus, Paal, and Sorgsepp?)

Leaving aside the question of holomorphy, do you generalize by picking a point p having a fiber isomorphic to

• (RxO)P^1 = OP^1 (the 8-sphere?)
• (CxO)P^1 (the 16-sphere?)
• (HxO)P^1 (the 32-sphere?)
• (OxO)P^1 (the 64-sphere?)

and then looking at what the topological structure of the candidate generalizations, to see what might be the corresponding structure to the Chern class of CP^2 ?

For instance, as stated in the paper by Iliev and Manivel at http://xxx.lanl.gov/abs/math.AG/0306329

"... the real Cayley plane F4 / Spin(9) ... admit[s] a cell decomposition R^0 u R^8 u R^16 and is topologically much simpler ...[than]... the complex Cayley plane ... E6 / P6 [a subset of] PJ3(O) ... ... 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 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 ...".

As to what might be a natural exceptional divisor for the case of the complexified octonionic projective plane (CxO)P^2 = E6 / Spin(10)xU(1), perhaps the paper of Landsberg and Manivel at http://xxx.lanl.gov/abs/math.AG/9908039 might be relevant, especially on pages 2, 24, and 25 where they say:

"... Let A denote ... the complexification of R, C, the quaternions H, or the octonions O ... ... Desingularizations ... Orbits in P(J3(A)) ... Let G / P = AP^2 [a subset of] PV be a Severi variety ... There is a natural diagram
```E = Q~^m -->   AP^2
in          in
PM    -f->  sigma(AP^2)
pi\|/
AP^2*```

where f is a desingularization of sigma(AP^2). The exceptional divisor E of f is natrually identified with ... the set of points in (AP^2*)* tangent to AP^2* along a quadric AP^1 =~ Q^m. ... we indicate the nodes defining the space of F-points AP^2 with black dots * and those defining the F-lines AP^2* =~ AP^2 with stars x. The bundle on AP^2* ... and the quadric ... are below the diagram...

```*--o--o--o--x
|
o
C^10
Q^8          ...".```

Notation and details are described in the paper. As of now, I don't know much about generalized del Pezzo surfaces beyond what I have written above, but I hope that some day I will find out more about the topological structure of the cases of

• (HxO)P^2 = E7 / Spin(12)xSp(1) (64-dim "quaternification")
• (OxO)P^2 = E8 / Spin(16) (128-dim "octonification").

Andy Neitzke also wrote: "... And anyway, these exceptional structures should be good for "something", right? ...".

I agree, but I will say that instead of working along the lines of conventional string and M-theory, I am working along the lines of my paper at http://xxx.lanl.gov/abs/physics/0207095 which uses exceptional structures and Clifford algebras in ways that are not widely accepted. ...".

"... I am not sure that I understand exactly what is needed for M-theory, but perhaps the Freudenthal-Tits magic squares as described in some of John Baez's This Weeks Finds might be relevant, as might be papers of Landsberg and Manivel in http://xxx.lanl.gov/abs/math.AG/9908039 and http://xxx.lanl.gov/abs/math.AG/0203260 where they give charts (similar to this)
```     R         C        H        O

R  v2(Q1)    P(TP2)   Gw(2,6)   OP2_C,0  hyperplane section of Severi
C  v2(P2)   P2 x P2    G(2,6)   OP2_C    Severi
H  Gw(3,6)   G(3,6)     S_12    E7/P7    lines through a point of G^ad

of varieties which are homogeneous spaces of Lie groups. Perhaps the fourth column

```OP2_C,0
OP2_C
E7/P7

might be relevant.

The first-above-cited paper has a section on Desingularizations.

The second-above-cited paper has a related diagram

```    R    C          H       O

R  RP2  CP2        HP2     OP2
C  CP2  CP2xCP2  GC(2,6)   OP2_C
H  HP2  GC(2,6)  GR(4,12)  E7(-5)
O  OP2  OP2_C     E7(-5)   E8(8)```

as to which they say "... The symmetric space E8(8) = E8 / SO(16) ... of dimension 128, is particularly intriguing and it would be very nice to have a direct construction of it. It is claimed in ... Rosenfeld ... Geomety of Lie Groups, ... Kluwer 1997 ... that it can be interpreted as a projective plane over OxO, and that in fact the whole square above of symmetric spaces can be obtained by taking projective planes, in a suitable sense, over the tensor products AxB. ...".

The first paper is Severi Varieties and their Varieties of Reductions and the second paper is The Chow Ring of the [complex] Cayley Plane, which can be regared as a complexification of the "real" OP2 = F4 / Spin(9) and has symmetric space structure E6 / Spin(10)xU(1) ...".

```
Many thanks to Douglas Welty for telling me about
the works of Charles Muses, and for sending me some
of them (they are hard to find).

References:

Charles Muses, The First Nondistributive Algebra,
with Relations to Optimization and Control Theory,
in Functional Analysis and Optimization,
ed. by E. R. Caianiello, Academic Press 1966.

Charles Muses, The Amazing 24th Dimension,
Journal for the Study of Consciousness, Research Notes.

Charles Muses, Hypernumbers and Quantum Field Theory
with a Summary of Physically Applicable Hypernumber
Arithmetics and their Geometries,
Applied Mathematics and Computation 6 (1960) 63-94.

Charles Muses, A Celebration of Higher Dimensional
Systems and a Man who Notably Explored Them,
Kybernetes 25 (1996) 48-52   (a review of
Kaleidoscopes, Selected Writings of H. S. M. Coxeter,
ed. by F. A. Sherk, P. McMullen, A. C. Thompson,
and A. I. Weiss, Wiley-Interscience 1995).

Kevin Carmody, Circular and Hyperbolic Quaternions,
Octonions, and Sedenions,
Applied Mathematics and Computation 28 (1988) 47-72.

Guillermo Moreno, The zero divisors of the Cayley-Dickson
algebras over the real numbers, q-alg/9710013.

R. Penrose and W. Rindler, Spinors and Space-Time,
vol. 2, Cambridge 1986,
particularly Appendix: spinors in n dimensions.

William Fulton and Joe Harris, Representation Theory,
Springer-Verlag 1991.

Boris Rosenfeld, Geometry of Lie Groups, Kluwer 1997.

Arthur L. Besse, Einstein Manifolds, Springer-Verlag 1987.

H. S. M. Coxeter, Regular Polytopes, 3rd ed, Dover 1973.

H. S. M. Coxeter, Integral Cayley Numbers,
Duke Math. J. 13 (1946) 561-578.

Kaleidoscopes, Selected Writings of H. S. M. Coxeter,
ed. by F. A. Sherk, P. McMullen, A. C. Thompson,
and A. I. Weiss, Wiley-Interscience 1995.
(Writings include Regular and Semi-Regular Polytopes, II,
Math. Zeitschrift 188 (1985) 559-591
and Regular and Semi-Regular Polytopes, III,
Math. Zeitschrift 200 (1988) 3-45
as well as many other papers.)
```