Conjugation Matters

Bioctonionic Veronese Vectors and Cayley-Rosenfeld Planes

Daniele Corradetti1, Alessio Marrani2, David Chester3 and Ray Aschheim3

1Departamento de Matemática, Universidade do Algarve,
Campus de Gambelas, 8005-139 Faro, Portugal
<EMAIL_ADDRESS>

2Instituto de Física Teorica, Dep.to de Física,
Universidad de Murcia, Campus de Espinardo, E-30100, Spain
<EMAIL_ADDRESS>

3Quantum Gravity Research,
Los Angeles, California, CA 90290, USA
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

Motivated by the recent interest in Lie algebraic and geometric structures
arising from tensor products of division algebras and their relevance to high
energy theoretical physics, we analyze generalized bioctonionic projective and
hyperbolic planes. After giving a Veronese representation of the
complexification of the Cayley plane $\mathbb{O}P_{\mathbb{C}}^{2}$, we
present a novel, explicit construction of the bioctonionic Cayley-Rosenfeld
plane $\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$, again by exploiting
Veronese coordinates. We discuss the isometry groups of all generalized
bioctonionic planes, recovering all complex and real forms of the exceptional
groups $F_{4}$ and $E_{6}$, and characterizing such planes as symmetric and
Hermitian symmetric spaces. We conclude by discussing some possible physical
applications.

###### Contents

  1. 1 Introduction
  2. 2 The Algebra of Bioctonions
    1. 2.1 Complex Norm
    2. 2.2 Real Norm
    3. 2.3 Automorphisms
  3. 3 Veronese Vectors over Bioctonions
    1. 3.1 The Complexification of the Cayley Plane $\mathbb{O}P_{\mathbb{C}}^{2}$
      1. 3.1.1 Complexified Cayley Lines
      2. 3.1.2 Elliptic and Hyperbolic Polarity on $\mathbb{O}P_{\mathbb{C}}^{2}$
      3. 3.1.3 Complexified Octonionic Affine Plane
    2. 3.2 The Bioctonionic Rosenfeld Plane $\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$
      1. 3.2.1 Rosenfeld Lines
      2. 3.2.2 Elliptic and Hyperbolic Polarity on $\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}P^{2}$
  4. 4 Veronese Vectors and Jordan Algebras
    1. 4.1 $\mathbb{O}P_{\mathbb{C}}^{2}$
    2. 4.2 $\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)P}^{2}$
  5. 5 Real Forms of $F_{4}$ and $E_{6}$
  6. 6 Bioctonionic Planes as Symmetric Spaces
  7. 7 Musings on the Physics of $E_{6(-78)}$ and $E_{6(2)}$
  8. 8 Conclusion

##  1 Introduction

Recently, attention has been focused by theoretical physicists on Lie
structures arising from tensor products of division algebras, and on their
geometrical characterizations. It is well known that Tits-Freudenthal magic
square has numerous applications in super Yang-Mills and supergravity theories
[GST, BM, CCM]. At the same time, extensions of the Standard Model based on
non-division algebras resulting from the Cayley-Dickson and their tensor
product was proposed in a recent article by Masi [Ma]. On a different, but
convergent, path, Todorov, Dubois-Violette [TDV] and Krasnov [Kra]
characterized the Standard Model gauge group $G_{SM}$ as a subgroup of the
automorphism group of the exceptional Jordan algebra
$\mathfrak{J}$${}_{3}\left(\mathbb{O}\right)$, while Boyle [Bo, Co] pointed to
its complexification $\mathfrak{J}_{3}^{\mathbb{C}}\left(\mathbb{O}\right)$.
It is well known that the group of determinant preserving automorphisms (also
named reduced structure group) of the complexification of the exceptional
Jordan algebra is related to the collineations of the complexification of the
Cayley plane $\mathbb{\mathbb{O}}P_{\mathbb{C}}^{2}$ (namely, the
complexification of the octonionic projective plane) and to
$E_{6}\left(\mathbb{C}\right)$, while the group of trace preserving
automorphisms of the same Jordan algebra is related with the isometries of the
Cayley plane and to $F_{4}\left(\mathbb{C}\right)$. While $F_{4}$ does not
have complex representations, the exceptional Lie Group $E_{6}$ is
historically well known as a candidate for Grand Unification Theories [GRS].

Complex and real forms of $E_{6}$ are also notoriously related to symmetries
of the bioctonionic Rosenfeld plane
$\mathbb{\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}}P^{2}$. In a series
of seminal papers resumed and completed in [Ros97, Ros98], Rosenfeld linked
all the Lie groups arising from Tits-Freudenthal magic square with the groups
of collineations and elliptic motions of generalised projective planes defined
over tensor product of division algebras and their split versions.
Consequently, he identified what are now known as Rosenfeld planes as
symmetric spaces and Hermitian symmetric spaces over compact and non compact
form of real Lie groups. However, in his articles Rosenfeld left some
ambiguity in the definition of the octonionic part of the construction : due
to the lack of associativity of octonions, the usual identification of the
plane through the quotient of a module could not indeed be pursued; moreover,
when the resulting tensor algebra is not a composition algebra, neither a
direct completion of a generalised affine plane could be worked out. A well
known identification, due to Jordan, von Neumann and Wigner [JvNW], relates
points of the projective plane $\mathbb{K}P^{2}$ with rank-1 idempotent
elements of the Jordan algebra $\mathfrak{J}_{3}\left(\mathbb{K}\right)$ of
Hermitian three by three matrices over the division algebra $\mathbb{K}$. This
provides a generalisation, but when defining a Jordan algebra is not possible,
the construction breaks down (see e.g. the discussion at the end of Sec. 5 of
[ABDN], and Refs. therein).

This paper is devoted to highlight the crucial role played by conjugation (and
related norm) in determining the algebraic-geometric structure of projective
plane (in a generalized sense) over tensor products of division algebras. In
the present paper we will focus on the bioctonionic algebra
$\mathbb{C}\otimes\mathbb{O}$, pointing out how its composition nature
strictly depends on the conjugation being considered. For the first time, we
will employ Veronese coordinates over the bioctonions in order to describe
suitable real forms of the bioctonionic Rosenfeld plane; this will pave the
way to the treatment of more complicated generalized projective spaces, which
we plan to deal with in future works. We will present an alternative, simple
construction, based on the definition of Veronese vectors over the
bioctonions, that allows an explicit description of two generalised projective
planes over the algebra of bioctonions $\mathbb{C}\otimes\mathbb{O}$ that are
of the most interest: the complexification of the octonionic projective plane
or Cayley plane $\mathbb{\mathbb{O}}P_{\mathbb{C}}^{2}$, and the bioctonionic
Rosenfeld plane
$\mathbb{\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}}P^{2}$ . In Sec. 2
we introduce the algebra of bioctonions with its linear structure,
conjugations and norms over both fields of real numbers $\mathbb{R}$ and
complex numbers $\mathbb{C}$. In Sec. 3 we explicitly construct
$\mathbb{\mathbb{O}}P_{\mathbb{C}}^{2}$ and
$\mathbb{\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}}P^{2}$ making use
of Veronese coordinates, while in the former case the construction has already
appeared in the literature (see e.g. [SBG11]), in the latter case the
construction by means of Veronese vectors is new, and it exhibits some non-
trivial features. Indeed, on one hand the complexification of the Cayley plane
$\mathbb{\mathbb{O}}P_{\mathbb{C}}^{2}$ is derived from a composition algebra,
it respects the Moufang identities and can be considered as a completion of a
generalised affine plane over the bioctonionic algebra. On the other hand,
since bioctonions are not a composition algebra with respect to a suitably
defined real norm, the bioctonionic Rosenfeld plane
$\mathbb{\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}}P^{2}$ violates the
basic axioms of projective geometry, and it cannot be considered as an
extension and completion of a would-be affine Rosenfeld plane. Then, in Sec. 4
we exploit the relation between Veronese vectors and simple, rank-3 Jordan
algebras, and we thus identify $\mathbb{\mathbb{O}}P_{\mathbb{C}}^{2}$ with
the space of rank-1 idempotent elements the complexification of the
exceptional Jordan Algebra
$\mathfrak{J}_{3}^{\mathbb{C}}\left(\mathbb{O}\right)$. In Sec. 5 we proceed
to analyze the group of motions of the generalised projective planes,
recovering $F_{4}$ as the isometry group of complexification of the Cayley
plane $\mathbb{\mathbb{O}}P_{\mathbb{C}}^{2}$, and $E_{6}$ as the isometry
group of the bioctonionic Rosenfeld plane
$\mathbb{\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}}P^{2}$. Sec. 6 then
deals with a systematic definition of bioctonionic planes as symmetric and
Hermitian symmetric spaces, retrieving all the real forms of $F_{4}$ and
$E_{6}$. Finally, in Sec. 7 we discuss some possible applications of $F_{4}$
and $E_{6}$ and the above related geometrical structures to high energy
theoretical physics. An outlook and comments on further future developments
are given in Sec. 8, which concludes the paper.

##  2 The Algebra of Bioctonions

Figure 1: Multiplication rule of octonions $\mathbb{O}$ (left) and of split-
octonions $\mathbb{O}_{s}$ (right) as real vector space $\mathbb{R}^{8}$ in
the basis $\left\\{i_{0}=1,i_{1},...,i_{7}\right\\}$. In the case of the
octonions $i_{0}^{2}=1$ and $i_{k}^{2}=-1$ for $k=1...7$, while in the case of
split-octonions $i_{k}^{2}=1$, for $k=1,2,3$ and $i_{k}^{2}=-1$ for $k\neq
1,2,3$. By exchanging the indices $2\leftrightarrow 3$ and $4\leftrightarrow
5$, the octonionic multiplication corresponds to the Cayley-Graves’ one [Cay,
Gra], recently discussed in [GKLY].

Let the octonions $\mathbb{O}$ be the only non-associative normed division
algebra with $\mathbb{O}_{s}$ as its split version, and let $\mathbb{C}$ be
the algebra of complex numbers and $\mathbb{C}_{s}$ its split algebra. We then
define the algebra of bioctonions as the complexification of the algebra of
Octonions, i.e. as the tensor product $\mathbb{C}\otimes\mathbb{O}$ or,
equivalently, as $\mathbb{C}\otimes\mathbb{O}_{s}$. Since $\mathbb{O}$ is an
alternative algebra and $\mathbb{C}$ is a commutative algebra, then
$\mathbb{C}\otimes\mathbb{O}$ is an alternative algebra. In the following
sections we will work with the $\mathbb{R}^{16}$ and the $\mathbb{C}^{8}$
decomposition of bioctonions. In the $\mathbb{R}^{16}$ decomposition, an
element of the bioctonionic algebra is given by

$b:={\sum_{\alpha=0}^{7}}\left(x^{\alpha}+\text{i}y^{\alpha}\right)i_{\alpha},$
(2.1)

where $x^{\alpha},y^{\alpha}\in\mathbb{R}$, the imaginary unit commutes with
the octonionic units , i.e. $\text{i}i_{\alpha}=i_{\alpha}\text{i}$, the
multiplication rules of $i_{\alpha}$ are given by the Fano plane in Fig.1 left
or right if bioctonions are considered as $\mathbb{C}\otimes\mathbb{O}$ or as
$\mathbb{C}\otimes\mathbb{O}_{s}$ respectively. Rewriting (2.1) we obtain the
$\mathbb{C}^{8}$ decomposition

$b:=\sum_{\alpha=0}^{7}z^{\alpha}i_{\alpha},$ (2.2)

where $z^{\alpha}\in\mathbb{C}$. The two decompositions (2.1) and (2.2)
highlight two different vector space structures available on the algebra of
bioctonions: the first is over the field of the real numbers $\mathbb{R}$ and
is of dimension 16, while the second is over the complex field $\mathbb{C}$
and has complex dimension $8$. It is worth noting that the algebra of
bioctonions is not a division algebra111The treatment of the Veronese vectors
over algebras containing zero-divisors has been given e.g. in [Cha], in which
a suitably generalized Veronese map is proposed (see Th. 5.2 therein)., e.g.
$\left(\text{i}i_{\alpha}+1\right)\left(\text{i}i_{\alpha}-1\right)=0$.

###  2.1 Complex Norm

Considering the bioctonions $\mathbb{C}\otimes\mathbb{O}$ as a complex vector
space, it is natural to define a complex norm. Let $b=z\otimes w$ a bioctonion
with $z\in\mathbb{C}$ and $w\in\mathbb{O}$, then its octonionic conjugate
$b^{\ast}$ is the element $b^{\ast}=z\otimes w^{\ast}$, where
$w^{\ast}\in\mathbb{O}$ is the conjugate of $\mathbb{O}$. Applying the complex
decomposition (2.1), then the octonionic conjugate of $b$ has the form

$b^{\ast}:=z^{0}-{\sum_{\alpha=1}^{7}}z^{\alpha}i_{\alpha}.$ (2.3)

We then define an octonionic inner product
$\left\langle\cdot,\cdot\right\rangle_{\mathbb{O}}$ over
$\mathbb{C}\otimes\mathbb{O}$ as

$\left\langle
b_{1},b_{2}\right\rangle_{\mathbb{O}}:=z_{1}^{0}z_{2}^{0}+...+z_{1}^{7}z_{2}^{7}\in\mathbb{C},$
(2.4)

where $z_{1}^{\alpha},z_{2}^{\alpha}\in\mathbb{C}$ are the complex
coefficients of $b_{1}$ and $b_{2}$ respectively. The octonionic inner product
induces a complex norm $N\left(\cdot\right)$ in $\mathbb{C}$, given as

$N\left(b\right):=\left\langle b,b\right\rangle_{\mathbb{O}}\in\mathbb{C},$
(2.5)

i.e.
$N\left(b\right)=\left(z^{0}\right)^{2}+...+\left(z^{7}\right)^{2}=bb^{\ast}=b^{\ast}b$.
The complex norm $N$ is a non degenerate quadratic form over the complex
vector space $\mathbb{C}\otimes\mathbb{O}$. Moreover, $b$ is a zero divisor if
and only if $N\left(b\right)=0$. In respect to the complex norm $N$ we also
have $N\left(\lambda b\right)=\lambda^{2}N\left(b\right)$ for every
$\lambda\in\mathbb{C}$, and

$N\left(b_{1}b_{2}\right)=N\left(b_{1}\right)N\left(b_{2}\right),$ (2.6)

and therefore $\mathbb{C}\otimes\mathbb{O}$ is a composition algebra with
respect to the complex norm $N$.

Remark 1. As the octonionic inner product induces an inner product and a
complex norm over $\mathbb{C}\otimes\mathbb{O}$, also its split version gives
rise to a split-octonionic inner product with a norm. Even though
$\mathbb{C}\otimes\mathbb{O}$ and $\mathbb{C}\otimes\mathbb{O}_{s}$ give rise
to the same bioctonionic algebra, we will write
$\mathbb{C}\otimes\mathbb{O}_{s}$ when we will intend the bioctonionic algebra
equipped with the split octonionic inner product and its norm.

###  2.2 Real Norm

We also define a real norm given by the bioctonionic conjugation, i.e.
$\overline{b}^{\ast}=\overline{z}\otimes w^{\ast}$, where $w^{\ast}$ is the
octonionic conjugate of $w$ in $\mathbb{O}$ and $\overline{z}$ is the complex
conjugate of $z$ in $\mathbb{C}$. Consequently the inner product
$\left\langle\cdot,\cdot\right\rangle_{\mathbb{C}\otimes\mathbb{O}}$ is
defined as

$\left\langle
b_{1},b_{2}\right\rangle_{\mathbb{C}\otimes\mathbb{O}}:=\overline{z}_{1}^{0}z_{2}^{0}+...+\overline{z}_{1}^{7}z_{2}^{7},$
(2.7)

and induce a real norm $\left\|\cdot\right\|$ that is the sum of the norms of
the complex coefficients of $b$, i.e.

$\left\|b\right\|^{2}:=\left\langle
b,b\right\rangle_{\mathbb{C}\otimes\mathbb{O}}=\left|z^{1}\right|^{2}+...+\left|z^{7}\right|^{2}\in\mathbb{R}.$
(2.8)

Since $\left\|ab\right\|^{2}\neq\left\|a\right\|^{2}\left\|b\right\|^{2}$ then
$\mathbb{C}\otimes\mathbb{O}$ is not a composition algebra __ in respect to
the real norm.

###  2.3 Automorphisms

Since the automorphisms of $\mathbb{C}$ are isomorphic to222We discard the so-
called ‘wild’ automorphisms [Ya]. $\mathbb{Z}_{2}$ [Bae], and the
automorphisms of $\mathbb{O}$ are isomorphic to the exceptional Lie group
$G_{2}$, i.e. $\text{Aut}\left(\mathbb{O}\right)\cong G_{2},$ the group of
automorphisms of the algebra of bioctonions is isomorphic to

$\text{Aut}\left(\mathbb{C}\otimes\mathbb{O}\right)=\mathbb{Z}_{2}\times
G_{2},$ (2.9)

and consequently the Lie algebra of derivations is isomorphic to that of
octonions, i.e.
$\mathfrak{der}\left(\mathbb{C}\otimes\mathbb{O}\right)\cong\mathfrak{g}_{2}$.

##  3 Veronese Vectors over Bioctonions

In this section we define explicitly two bioctonionic planes making use of
Veronese coordinates. In the case of the projective and hyperbolic planes on
octonions, i.e. $\mathbb{O}P^{2}$ and $\mathbb{O}H^{2}$ respectively, the
construction is well known [SBG11]. The projective plane over the octonions
with real coefficients
$\mathbb{R}\otimes\mathbb{O}\equiv\mathbb{O}_{\mathbb{R}}$ has been studied
extensively in [Tit53, Fre54]. A rigorous definition of the octonionic
projective plane, and the proof that its automorphism group is a simple group
of type $E_{6}$ in all characteristics, can be found in [Cha] (see Th. 5.1
therein).

In the present paper, we study the complexification of the Cayley Plane
$OP_{\mathbb{C}}^{2}$ and the bioctonionic Rosenfeld plane
$\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$. In the former case, we make
use of the complex vector space structure of $\mathbb{C}\otimes\mathbb{O}$,
and therefore of the octonionic conjugation $b^{\ast}$, the octonionic inner
product $\left\langle\cdot,\cdot\right\rangle_{\mathbb{O}}$ and complex norm
$N\left(\cdot\right)$. In the latter case, we rely on the real vector space
structure of $\mathbb{C}\otimes\mathbb{O}$, and therefore on the bioctonionic
conjugation $\overline{b}^{\ast}$, the bioctonionic inner product
$\left\langle\cdot,\cdot\right\rangle_{\mathbb{\mathbb{C}\otimes\mathbb{O}}}$
and real norm $\left\|\cdot\right\|$; we will provide a Veronese
representation of the bioctonionic Rosenfeld plane
$\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$ even if the bioctonions are
not a composition algebra with respect to the real norm $\left\|\cdot\right\|$

###  3.1 The Complexification of the Cayley Plane
$\mathbb{O}P_{\mathbb{C}}^{2}$

Let
$V\cong\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}^{3}\times\mathbb{C}^{3}$
be a complex vector space, with elements $\omega$ of the form

$\left(b_{\nu};\lambda_{\nu}\right)_{\nu}=\left(b_{1},b_{2},b_{3};\lambda_{1},\lambda_{2},\lambda_{3}\right),$
(3.1)

where $b_{\nu}\in\mathbb{\mathbb{C}\otimes\mathbb{O}}$,
$\lambda_{\nu}\in\mathbb{C}$ for $\nu=1,2,3$. A vector $\omega\in V$ is called
Veronese iff

$\displaystyle\lambda_{1}b_{1}^{\ast}$
$\displaystyle=b_{2}b_{3},\,\,\lambda_{2}b_{2}^{\ast}=b_{3}b_{1},\,\,\lambda_{3}b_{3}^{\ast}=b_{1}b_{2},$
(3.2) $\displaystyle N\left(b_{1}\right)$
$\displaystyle=\lambda_{2}\lambda_{3},\,N\left(b_{2}\right)=\lambda_{3}\lambda_{1},N\left(b_{1}\right)=\lambda_{1}\lambda_{2}.$
(3.3)

Let the set $H\subset V$ be the set of Veronese vectors inside $V$. Since
$\mathbb{C}$ is commutative and $\lambda^{\ast}=\lambda$, since $\lambda
b=b\lambda$ and $N\left(\lambda b\right)=\lambda^{2}N\left(b\right)$ when
$\lambda\in\mathbb{C}$, then if $\omega$ is a Veronese vector, all complex
multiples $\mathbb{C\omega}$ are again Veronese vectors, i.e. if $\omega\in H$
then $\mu\omega\in H$ when $\mu\in\mathbb{C}$. We then define the complexified
octonionic plane or complexification of the Cayley plane
$\mathbb{O}P_{\mathbb{C}}^{2}$ as the set of 1-dimensional complex subspaces
$\mathbb{C}\omega$ that we will call points of the plane, i.e.

$\mathbb{O}P_{\mathbb{C}}^{2}:=\left\\{\mathbb{C}\omega:\omega\in
H\smallsetminus\left\\{0\right\\}\right\\}.$ (3.4)

####  3.1.1 Complexified Cayley Lines

Lines in $\mathbb{O}P_{\mathbb{C}}^{2}$ are orthogonal subspaces of a point of
the plane. Therefore, let $\mathbb{C}\omega$ be a point in
$\mathbb{O}P_{\mathbb{C}}^{2}$, we define the line $\ell$ as the orthogonal
subspace

$\ell:=\omega^{\perp}=\left\\{\upsilon\in
V:\beta\left(\upsilon,\omega\right)=0\right\\},$ (3.5)

where the complex bilinear form $\beta$ is given by

$\beta\left(\upsilon,\omega\right):={\sum_{\nu=1}^{3}}\left(\left\langle
b_{\nu}^{1},b_{\nu}^{2}\right\rangle_{\mathbb{O}}+\lambda_{\nu}^{1}\lambda_{\nu}^{2}\right),$
(3.6)

with $\upsilon,\omega\in V$, of coordinates
$\left(b_{\nu}^{1};\lambda_{\nu}^{1}\right)_{\nu}$,$\left(b_{\nu}^{2};\lambda_{\nu}^{2}\right)_{\nu}$
respectively.

####  3.1.2 Elliptic and Hyperbolic Polarity on $\mathbb{O}P_{\mathbb{C}}^{2}$

Since every point $\mathbb{C}\omega$ of the plane defines an orthogonal line
$\omega^{\perp}\subset\mathbb{O}P_{\mathbb{C}}^{2}$ and, as converse, every
line defines a point, we call standard elliptic polarity $\pi^{+}$ the
involutive map that corresponds points to lines and lines to points through
orthogonality, i.e.

$\pi^{+}\left(\omega\right):=\omega^{\perp},\pi^{+}\left(\omega^{\perp}\right):=\omega,$
(3.7)

using the complex bilinear form $\beta\left(\cdot,\cdot\right)$ so that

$\omega\longrightarrow\left\\{\beta\left(\cdot,\omega\right)=0\right\\}.$
(3.8)

Explicitly, $\beta\left(\upsilon,\omega\right)=0$ when

$b_{1}^{1}b_{1}^{2\ast}+b_{2}^{1}b_{2}^{2\ast}+b_{3}^{1}b_{3}^{2\ast}+\lambda_{1}^{1}\lambda_{1}^{2}+\lambda_{2}^{1}\lambda_{2}^{2}+\lambda_{3}^{1}\lambda_{3}^{2}=0,$
(3.9)

where, as before, we intended,
$\left(b_{\nu}^{1};\lambda_{\nu}^{1}\right)_{\nu}$ and
$\left(b_{\nu}^{2};\lambda_{\nu}^{2}\right)_{\nu}$ as the coordinates of
$\upsilon,\omega\in V$. We also define an hyperbolic polarity $\pi^{-}$ as the
involutive map between points and lines which still has

$\pi^{-}\left(\omega\right):=\omega^{\perp},\pi^{-}\left(\omega^{\perp}\right):=\omega,$
(3.10)

but through the use of the bilinear form $\beta_{-}$ which has a change of
sign in the last coordinate, i.e. $\beta_{-}\left(\upsilon,\omega\right)=0$
when

$b_{1}^{1}b_{1}^{2\ast}+b_{2}^{1}b_{2}^{2\ast}-b_{3}^{1}b_{3}^{2\ast}+\lambda_{1}^{1}\lambda_{1}^{2}+\lambda_{2}^{1}\lambda_{2}^{2}-\lambda_{3}^{1}\lambda_{3}^{2}=0.$
(3.11)

The projective plane equipped with the bilinear form $\beta_{-}$ and the
hyperbolic polarity __ $\pi^{-}$ it will be called the complexified hyperbolic
Cayley plane $\mathbb{O}H_{\mathbb{C}}^{2}$.

####  3.1.3 Complexified Octonionic Affine Plane

Figure 2: Representation of the affine plane: $\left(0,0\right)$ represents
the origin, $\left(0\right)$ the point at the infinity on the $x$-axis,
$\left(s\right)$ is the point at infinity of the line $\left[s,t\right]$ of
slope $s$ while $\left(\infty\right)$ is the point at the infinity on the
$y$-axis and of vertical lines $\left[c\right]$.

In analogy to the classic case, the complexification of the Cayley plane can
also be seen as the completion and topological compactification of a
bioctonionic affine plane $\left(\mathbb{C}\otimes\mathbb{O}\right)A^{2}$,
but, since $\mathbb{C}\otimes\mathbb{O}$ is not a division algebra, the strict
set of axioms and results of affine geometry are not valid on this plane.
Indeed, the map from $\left(\mathbb{C}\otimes\mathbb{O}\right)^{2}$ to $V$
defined as

$\left(x,y\right)\mapsto\mathbb{C}\left(x,y^{\ast},yx^{\ast};N\left(y\right),N\left(x\right),1\right),$
(3.12)

sends elements
$\left(x,y\right)\in\left(\mathbb{C}\otimes\mathbb{O}\right)^{2}$ into
Veronese vectors, and therefore to $\mathbb{O}P_{\mathbb{C}}^{2}$ establishing
a correspondence between points of the complexification of the Cayley plane
and elements of an affine plane
$\left(\mathbb{C}\otimes\mathbb{O}\right)A^{2}$ with bioctonionic coordinates
$\left(x,y\right)$. To show that $\mathbb{O}P_{\mathbb{C}}^{2}$ is a
completion of the affine plane we add two sets of point of coordinates
$\left(x\right)$ and $\left(\infty\right)$ that will extend the map to cover
the whole $\mathbb{O}P_{\mathbb{C}}^{2}$ as follows :

$\displaystyle\left(x\right)$
$\displaystyle\mapsto\mathbb{C}\left(0,0,x;N\left(x\right),1,0\right),$ (3.13)
$\displaystyle\left(\infty\right)$
$\displaystyle\mapsto\left(0,0,0;1,0,0\right),$ (3.14)

where $x\in\mathbb{C}\otimes\mathbb{O}$. Finally, in order to give a complete
picture, a line in the affine plane
$\left(\mathbb{C}\otimes\mathbb{O}\right)A^{2}$ will be given by

$\left[s,t\right]:=\left\\{\left(x,sx+t\right):x,t\in\mathbb{\mathbb{C}\otimes\mathbb{O}}\right\\},$
(3.15)

where $s$ $\in\mathbb{C}\otimes\mathbb{O}$ is the slope of the line. Vertical
lines will be indicated as
$\left[c\right]:=\left\\{c\right\\}\times\left(\mathbb{C}\otimes\mathbb{O}\right)$.
As for the previous case, lines of the affine plane
$\left(\mathbb{C}\otimes\mathbb{O}\right)A^{2}$ have a correspondence with
lines of the projective plane $\mathbb{O}P_{\mathbb{C}}^{2}$ through the map

$\displaystyle\left[s,t\right]$
$\displaystyle\mapsto\mathbb{C}\left(s^{\ast}t,-t^{\ast},-s;1,N\left(s\right),N\left(t\right)\right)^{\perp},$
(3.16) $\displaystyle\left[c\right]$
$\displaystyle\mapsto\mathbb{C}\left(-c,0,0;0,1,N\left(c\right)\right)^{\perp},$
(3.17)

where $s,t,c\in\mathbb{C}\otimes\mathbb{O}$. The correspondence is bijective
if we add a line $\left[\infty\right]$ that will correspond to

$\left[\infty\right]\mapsto\mathbb{C}\left(0,0,0;0,0,1\right)^{\perp}.$ (3.18)

Remark 2. The complexification of the octonionic affine plane does not satisfy
the axioms of usual affine geometry. It is critical the existence of adjacent
points, i.e. points that are separated by a singular affine vector
$\left(v_{1},v_{2}\right)$ such that $(av_{1},av_{2})=0$, with
$a,v_{1},v_{2}\in\mathbb{C}\otimes\mathbb{O}$. Between two adjacent points
passes more than one line, that are called adjacent themselves. Two lines that
are not adjacent but that can be transformed into adjacent through a
translation are called diverging lines. So any two given lines in this plane
might be incident, parallel, coincident, adjacent, or divergent (cfr.
[Ros97]).

Remark 3. The counting of (complex) dimensions goes as follows. By
construction, dim${}_{\mathbb{C}}V=27$. Since $\mathbb{C}\otimes\mathbb{O}$ is
composition with respect to the complex norm $N\left(\cdot\right)$, only 10
relations are independent out of all 27 ones (3.2)-(3.3) defining Veronese
vectors; thus dim${}_{\mathbb{C}}H$ $=17$. Finally, from (3.4)
dim${}_{\mathbb{C}}(\mathbb{O}P_{\mathbb{C}}^{2})=$dim${}_{\mathbb{C}}H-1=16$,
as expected.

###  3.2 The Bioctonionic Rosenfeld Plane
$\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$

We now replicate the same construction, considering as a starting point the
real vector space
$V\cong\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}^{3}\times\mathbb{R}^{3}$
and the real norm over the bioctonionic algebra. In this case elements of $V$
are of the form

$\left(b_{\nu};\lambda_{\nu}\right)_{\nu}=\left(b_{1},b_{2},b_{3};\lambda_{1},\lambda_{2},\lambda_{3}\right),$
(3.19)

where $b_{\nu}\in\mathbb{\mathbb{C}\otimes\mathbb{O}}$,
$\lambda_{\nu}\in\mathbb{R}$ and $\nu=1,2,3$. A vector $\omega\in V$ is called
Veronese iff

$\displaystyle\lambda_{1}\overline{b}_{1}^{\ast}$
$\displaystyle=b_{2}b_{3},\,\,\lambda_{2}\overline{b}_{2}^{\ast}=b_{3}b_{1},\,\,\lambda_{3}\overline{b}_{3}^{\ast}=b_{1}b_{2},$
(3.20) $\displaystyle\left\|b_{1}\right\|^{2}$
$\displaystyle=\lambda_{2}\lambda_{3},\,\left\|b_{2}\right\|^{2}=\lambda_{3}\lambda_{1},\left\|b_{3}\right\|^{2}=\lambda_{1}\lambda_{2}.$
(3.21)

Let $H\subset V$ be the set of Veronese vectors. Since $\mathbb{R}$ is a
commutative field, $\mu^{\ast}=\mu$ and $\left\|\mu
b\right\|^{2}=\mu^{2}\left\|b\right\|^{2}$ when $\mu\in\mathbb{R}$, given a
Veronese vector $\omega\in H$, all real multiples $\mathbb{R\omega}$ are again
Veronese vectors, i.e. $\mathbb{R}\omega\subset H$. We define the bioctonionic
Rosenfeld plane $\mathbb{\left(\mathbb{C}\otimes O\right)}P^{2}$ as the set of
1-dimensional subspaces $\mathbb{R}\omega$ that we will call points of the
plane, i.e.

$\mathbb{\left(\mathbb{C}\otimes
O\right)}P^{2}:=\left\\{\mathbb{R}\omega:\omega\in
H\smallsetminus\left\\{0\right\\}\right\\}.$ (3.22)

####  3.2.1 Rosenfeld Lines

As in the case of the complexification of the Cayley plane, we define the
lines of the bioctonionic Rosenfeld plane $\mathbb{\left(\mathbb{C}\otimes
O\right)}P^{2}$ as orthogonal subspaces of a point through the extension of
the bioctonionic inner product. Therefore, let $\mathbb{R}\omega$ be a point
in $\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}P^{2}$, then the line
$\ell$ is the orthogonal subspace

$\ell:=\omega^{\perp}=\left\\{\upsilon\in
V:\beta_{\mathbb{C}\otimes\mathbb{O}}\left(\upsilon,\omega\right)=0\right\\},$
(3.23)

where the bilinear form is defined as

$\beta_{\mathbb{C}\otimes\mathbb{O}}\left(\upsilon,\omega\right):={\sum_{\nu=1}^{3}}\left(\left\langle
b_{\nu}^{1},b_{\nu}^{2}\right\rangle_{\mathbb{\mathbb{C}\otimes
O}}+\lambda_{\nu}^{1}\lambda_{\nu}^{2}\right),$ (3.24)

with $\upsilon,\omega\in V$ of coordinates
$\left(b_{\nu}^{1};\lambda_{\nu}^{1}\right)_{\nu}$,$\left(b_{\nu}^{2};\lambda_{\nu}^{2}\right)_{\nu}$
respectively.

####  3.2.2 Elliptic and Hyperbolic Polarity on
$\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}P^{2}$

Since every point $\mathbb{R}\omega$ defines an orthogonal line
$\omega^{\perp}\subset\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}P^{2}$
and every line defines a point, we call standard elliptic polarity $\pi^{+}$
the involutive map that correspond points to lines and lines to points through
orthogonality, i.e.

$\pi^{+}\left(\omega\right):=\omega^{\perp},\pi^{+}\left(\omega^{\perp}\right):=\omega,$
(3.25)

making use of the bilinear form
$\beta_{\mathbb{C}\otimes\mathbb{O}}\left(\cdot,\cdot\right)$. Explicitly,
$\beta_{\mathbb{C}\otimes\mathbb{O}}\left(\upsilon,\omega\right)=0$ when

$b_{1}^{1}\overline{b}_{1}^{2\ast}+b_{2}^{1}\overline{b}_{2}^{2\ast}+b_{3}^{1}\overline{b}_{3}^{2\ast}+\lambda_{1}^{1}\lambda_{1}^{2}+\lambda_{2}^{1}\lambda_{2}^{2}+\lambda_{3}^{1}\lambda_{3}^{2}=0,$
(3.26)

where, as before, we intended,
$\left(b_{\nu}^{1};\lambda_{\nu}^{1}\right)_{\nu}$ and
$\left(b_{\nu}^{2};\lambda_{\nu}^{2}\right)_{\nu}$ as the coordinates of
$\upsilon,\omega\in V$.

As in the previous case, the elliptic polarity is not the only possible one,
indeed we define the hyperbolic polarity $\pi^{-}$__ as the involutive map
between points and lines which still has

$\pi^{-}\left(\omega\right):=\omega^{\perp},\pi^{-}\left(\omega^{\perp}\right):=\omega,$
(3.27)

but that uses the bilinear form $\beta_{\mathbb{C}\otimes\mathbb{O}}^{-}$
which has a change of sign in the last coordinate, i.e.
$\beta_{\mathbb{C}\otimes\mathbb{O}}^{-}\left(\upsilon,\omega\right)=0$ when

$b_{1}^{1}\overline{b}_{1}^{2\ast}+b_{2}^{1}\overline{b}_{2}^{2\ast}-b_{3}^{1}\overline{b}_{3}^{2\ast}+\lambda_{1}^{1}\lambda_{1}^{2}+\lambda_{2}^{1}\lambda_{2}^{2}-\lambda_{3}^{1}\lambda_{3}^{2}=0.$
(3.28)

The projective plane equipped with the bilinear form
$\beta_{\mathbb{C}\otimes\mathbb{O}}^{-}$ instead of
$\beta_{\mathbb{C}\otimes\mathbb{O}}$ it will be called the bioctonionic
Rosenfeld hyperbolic plane
$\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}H^{2}$.

Remark 4. Since $\mathbb{\mathbb{C}\otimes\mathbb{O}}$ is not a composition
algebra with respect of the real norm $\left\|\cdot\right\|$, a map as in
(3.12) would not be well defined. Therefore, we cannot consider the
Bioctonionic Rosenfeld plane as an extension and completion of an affine
Rosenfeld plane.

Remark 5. The counting of (real) dimensions goes as follows. Since
$\mathbb{\mathbb{C}\otimes\mathbb{O}}$ is not a composition algebra with
respect to the real norm $\left\|\cdot\right\|$, one must consider (3.21) and,
say, the first of (3.20), as independent relations out of the relations
(3.20)-(3.21) defining Veronese vectors. This corresponds to $8\times 2$
$+1+1+1=19$ real conditions out of the $8\times 2\times 3+1+1+1=51$ real
relations (3.20)-(3.21). Thus, the real dimension of
$\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}P^{2}$ is given by
dim${}_{\mathbb{R}}\left(\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}P^{2}\right)=51-19=32$,
as expected.

##  4 Veronese Vectors and Jordan Algebras

A well known identification relates rank-1 idempotent elements of the Jordan
algebra $\mathfrak{J}_{3}\left(\mathbb{O}\right)$ with points of the
octonionic projective plane $\mathbb{O}P^{2}$. While this identification still
stands for the complexification of the Cayley plane
$\mathbb{O}P_{\mathbb{C}}^{2}$, when it is applied to the Rosenfeld plane
$\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}P^{2}$ one obtains that
$\mathfrak{J}_{3}\left(\mathbb{C}\otimes\mathbb{O}\right)$ is a simple Jordan
algebra, but not a formally real one (cfr. e.g. [Bae]).

As we discussed above, Veronese vectors, defined by conditions in (3.2), are
an alternative and useful way to characterize rank-1 idempotent elements of a
Jordan algebra $\mathfrak{J}_{3}\left(\mathbb{K}\right)$, with $\mathbb{K}$
being any tensor product of division algebras.

###  4.1 $\mathbb{O}P_{\mathbb{C}}^{2}$

In order to show such relation within $\mathbb{O}P_{\mathbb{C}}^{2}$, let
$\omega$ be an element of
$V\cong\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}^{3}\times\mathbb{C}^{3}$
with coordinates
$\left(b_{1},b_{2},b_{3};\lambda_{1},\lambda_{2},\lambda_{3}\right)$ and
define $A_{\omega}$ as the three by three bioctonionic matrix given by

$V\ni\omega\longrightarrow
A_{\omega}:=\left(\begin{array}[]{ccc}\lambda_{1}&b_{3}&b_{2}^{\ast}\\\
b_{3}^{\ast}&\lambda_{2}&b_{1}\\\
b_{2}&b_{1}^{\ast}&\lambda_{3}\end{array}\right)\in\mathbb{C}\otimes\mathfrak{J}_{3}\left(\mathbb{O}\right).$
(4.1)

Note that since the scalar field $\mathbb{C}$ commutes with the coefficient
$b$, then all $\mathbb{C}$-multiples of the vector $\omega$ are sent in
multiple of the matrix $A_{\omega}$. Therefore the map is well defined and
induces a bijective map between points
$\mathbb{C}\omega\in\mathbb{O}P_{\mathbb{C}}^{2}$ and subspaces of the form
$\mathbb{C}A_{\omega}$. It is here worth recalling that the cubic norm
$\mathcal{N}$ of a non-zero element
$A_{\omega}\in\mathbb{C}\otimes\mathfrak{J}_{3}\left(\mathbb{O}\right)$ is
defined in terms of the generalization of the determinant for three by three
matrices with not necessarily associative elements333See e.g. the example 5 of
[Krut], which actually is a simplified version of the reduced cubic factor
example in Sec. I.3.9 of [McCr]. :

$\mathcal{N}\left(A_{\omega}\right)\equiv\text{det}\left(A_{\omega}\right):=\lambda_{1}\lambda_{2}\lambda_{3}-\lambda_{1}N(b_{1})-\lambda_{2}N\left(b_{2}\right)-\lambda_{3}N\left(b_{3}\right)+2\text{Re}\left(\left(b_{1}b_{2}\right)b_{3}\right),$
(4.2)

where
Re$\left(\left(b_{1}b_{2}\right)b_{3}\right):=\frac{1}{2}\left(\left(b_{1}b_{2}\right)b_{3}+\overline{b_{3}}\left(\overline{b_{2}}\overline{b_{1}}\right)\right)$,
implying that det$\left(aA_{\omega}\right)=a^{3}$det$\left(A_{\omega}\right)$,
$\forall a\in\mathbb{C}$. It should be remarked that the determinant is
actually well defined, as one can realize by recalling the Hamilton Cayley
identity (see e.g. [Yok]), i.e.

$A_{\omega}\circ
A_{\omega}^{2}-\text{tr}\left(A_{\omega}\right)A_{\omega}^{2}+\frac{1}{2}\left(\text{tr}\left(A_{\omega}\right)^{2}-\text{tr}\left(A_{\omega}^{2}\right)\right)A_{\omega}=\text{det}\left(A_{\omega}\right)I,$
(4.3)

where $I$ is the three by three identity matrix, and $\circ$ is the Jordan
product $A\circ B:=\frac{1}{2}\left(AB+BA\right)$.

By further specifying that $\omega$ is a Veronese vector in $V$, namely that
$\omega\in H\subset V$, and thus by plugging the Veronese conditions
(3.2)-(3.3) into (4.2), one obtains that the norm of the corresponding element
$A_{\omega}$ vanishes :

$\omega\in H\Rightarrow\mathcal{N}\left(A_{\omega}\right)=0.$ (4.4)

Moreover, one can consider the image $A_{\omega}^{\sharp}$ of a non-zero
element $A_{\omega}$, with $\omega\in H$, under the so-called adjoint
($\sharp$-)map of $\mathbb{C}\otimes\mathfrak{J}_{3}\left(\mathbb{O}\right)$,
which is given by (again, cf. e.g. example 5 of [Krut])

$A_{\omega}^{\sharp}:=\left(\begin{array}[]{ccc}\lambda_{2}\lambda_{3}-N(b_{1})&\overline{b_{2}}\overline{b_{1}}-\lambda_{3}b_{3}&b_{3}b_{1}-\lambda_{2}\overline{b_{2}}\\\
b_{1}b_{2}-\lambda_{3}\overline{b_{3}}&\lambda_{1}\lambda_{3}-N(b_{2})&\overline{b_{3}}\overline{b_{2}}-\lambda_{1}b_{1}\\\
\overline{b_{1}}\overline{b_{3}}-\lambda_{2}b_{2}&b_{2}b_{3}-\lambda_{1}\overline{b_{1}}&\lambda_{1}\lambda_{2}-N(b_{3})\end{array}\right),$
(4.5)

and which, when recalling (3.2)-(3.3), can be realized to vanish (again!),
thus implying that

$\omega\in H\Leftrightarrow A_{\omega}^{\sharp}=0.$ (4.6)

Thus, by the Definition 11 of [Krut] (namely, from the invariant definition of
the rank of an element of $\mathfrak{J}_{3}^{\mathbb{O}}$ [Jac61]), one
obtains that the Veronese conditions (3.2)-(3.3) are an equivalent
characterization of the rank-1 elements of the complexification of the
exceptional Jordan algebra
$\mathbb{C}\otimes\mathfrak{J}_{3}\left(\mathbb{O}\right)\equiv\mathfrak{J}_{3}^{\mathbb{C}}\left(\mathbb{O}\right)$
(see e.g. [Jac68] for an extensive analysis).

Remark 6. The definition (3.4) characterizes the points of
$\mathbb{O}P_{\mathbb{C}}^{2}$ as complex ‘Veronese rays’, thus obtaining a
$16_{\mathbb{C}}$-dimensional subspace of the unique orbit of rank-1 elements
of $\mathfrak{J}_{3}^{\mathbb{C}}\left(\mathbb{O}\right)$. Any well defined
representative of such a 16-dimensional subspace has a fixed trace.

###  4.2 $\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)P}^{2}$

The same argument, with a different ending point, applies to the bioctonionic
Rosenfeld plane. Given a Veronese vector $\omega$ in
$V\cong\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}^{3}\times\mathbb{R}^{3}$
with coordinates
$\left(b_{1},b_{2},b_{3};\lambda_{1},\lambda_{2},\lambda_{3}\right)$ we define
$A_{\omega}$ as the three by three bioctonionic Hermitian matrix as

$\omega\longrightarrow
A_{\omega}:=\left(\begin{array}[]{ccc}\lambda_{1}&b_{3}&\overline{b}_{2}^{\ast}\\\
\overline{b}_{3}^{\ast}&\lambda_{2}&b_{1}\\\
b_{2}&\overline{b}_{1}^{\ast}&\lambda_{3}\end{array}\right).$ (4.7)

Since the scalar field $\mathbb{R}$ commutes with the coefficient $b$, then
all $\mathbb{R}$-multiples of the vector $\omega$ are sent in multiple of the
matrix $A_{\omega}$. Therefore, the map is well defined and induces a
injective map between points
$\mathbb{\mathbb{R}}\omega\in\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)P}^{2}$
and subspaces $\mathbb{\mathbb{R}}A_{\omega}\subset
H_{3}\left(\mathbb{C}\otimes\mathbb{O}\right)$, the algebra of $3\times 3$
matrices with $\left(\mathbb{C}\otimes\mathbb{O}\right)$-valued entries and
Hermitian with respect to the bi-octonionic conjugation; this time, the
bioctonionic conjugation still allows $A_{\omega}$ to be endowed with the
structure of a simple Jordan algebra, but not of a formally real one.

##  5 Real Forms of $F_{4}$ and $E_{6}$

Symmetries of the generalised projective planes over the bioctonionic algebra
lead naturally to all the complex and real forms of the exceptional group
$E_{6}$ and $F_{4}$. To do so we will look to generalised collineations, i.e.
automorphisms of generalised planes that sends lines into lines, and elliptic
and hyperbolic motions, i.e. collineations over the projective (hyperbolic)
plane that preserve the elliptic (hyperbolic) polarity. Sometimes, due to
correspondence between idempotent Jordan matrices and points in the projective
plane, the collineation group of the octonionic plane
$\text{Coll}\left(\mathbb{O}P^{2}\right)$ is called
$SL\left(3,\mathbb{O}\right)$, while the elliptic polarity preserving group
$\text{Iso}\left(\mathbb{O}P^{2}\right)$ is identified with
$SU\left(3,\mathbb{O}\right)$ [DW].

To recover the collineations group and the polarity preserving group of the
octonionic projective and hyperbolic space, one might proceed in a geometric
[SBG11], group algebraic [Jac68] and Lie algebraic way [Ros98]. We will
represent the last one following Rosenberg focusing on the Lie algebra of the
collineation group $\text{Coll}\left(\mathbb{O}P^{2}\right)$ that is given by
the direct sum the Lie Algebra given by the group of automorphisms of the
field, in this case $\text{Aut}\left(\mathbb{O}\right)=G_{2}$, and the algebra
$\mathfrak{a}_{3}\left(\mathbb{O}\right)$ of three by three matrices on
$\mathbb{O}$ and null trace, i.e. $\text{tr}\left(A\right)=0$. We therefore
have

$\mathfrak{coll}\left(\mathbb{O}P^{2}\right)=\mathfrak{g}_{2}\oplus\mathfrak{a}_{3}\left(\mathbb{O}\right).$
(5.1)

A simple count on the dimension on the generators of the
$\mathfrak{a}_{3}\left(\mathbb{O}\right)$ algebra, imposing the null trace
condition, leads $8$ entries of dimension $8$ and therefore
$\text{dim}{}_{\mathbb{R}}\mathfrak{a}_{3}=64$, that brings to

$\text{dim}_{\mathbb{R}}\left(\mathfrak{coll}\left(\mathbb{O}P^{2}\right)\right)\cong
78=64+14,$ (5.2)

which leads to the group $\text{Coll}\left(\mathbb{O}P^{2}\right)$ be a
$E_{6}$ type Lie group as expected.

The same argument is applied for the polarity preserving group
$\text{Iso}\left(\mathbb{O}P^{2}\right)$, i.e. collineations that preserve
also the elliptic polarity $\pi^{+}$ or equivalently the form $\beta$. This
argument leads to the Lie algebra $\mathfrak{iso}\left(\mathbb{O}P^{2}\right)$
that is given

$\mathfrak{iso}\left(\mathbb{O}P^{2}\right)=\mathfrak{g}_{2}\oplus\mathfrak{sa}_{3}\left(\mathbb{O}\right),$
(5.3)

where we intended $\mathfrak{sa}_{3}\left(\mathbb{O}\right)$ the anti-
Hermitian matrices, i.e. $a_{ij}=-a_{ji}^{\ast}$, of null trace. Elements of
this algebra are of the form

$A=\left(\begin{array}[]{ccc}a_{1}^{1}&a_{2}^{1}&-\left(a_{3}^{1}\right)^{\ast}\\\
-\left(a_{2}^{1}\right)^{\ast}&a_{2}^{2}&a_{3}^{2}\\\
a_{3}^{1}&-\left(a_{3}^{2}\right)^{\ast}&a_{3}^{3}\end{array}\right),$ (5.4)

with $a_{3}^{3}=-\left(a_{1}^{1}+a_{2}^{2}\right)$ and
$\text{Re}\left(a_{1}^{1}\right)=\text{Re}\left(a_{2}^{2}\right)=0$. The
dimension count on the generators of the algebra leads to 3 entries of
dimension $8$, $2$ of dimension $7$ and therefore
$\text{dim}_{\mathbb{R}}\mathfrak{sa}_{3}\left(\mathbb{O}\right)=38$ and
therefore

$\text{dim}_{\mathbb{R}}\mathfrak{iso}\left(\mathbb{O}P^{2}\right)\cong
52=38+14,$ (5.5)

which points to $\text{Iso}\left(\mathbb{O}P^{2}\right)$ as an $F_{4}$-type
Lie group as expected. With more efforts, following Yokota [Yok], we can
recover all isometry groups giving rise to complex and real forms of $F_{4}$
(Table 1).

Plane | Isometry group
---|---
$\mathbb{O}P^{2}\left(\mathbb{C}\right)$ | $F_{4}\left(\mathbb{C}\right)$
$\mathbb{O}P^{2}$ | $F_{4\left(-52\right)}$
$\mathbb{O}_{s}P^{2}$ | $F_{4\left(4\right)}$
$\mathbb{O}H^{2}$ | $F_{4\left(-20\right)}$
Table 1: The isometry group of the octonionic and split-octonionic projective
and hyperbolic planes give rise to complex and real forms of $F_{4}$.

The same argument can be applied to the elliptic motion group of the
bioctonionic Rosenfeld plane
$\text{Iso}\left(\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}P^{2}\right)$.
Since $\mathbb{C}$ and $\mathbb{O}$ are both composition algebras, with
$\mathfrak{der}\left(\mathbb{C}\right)\cong 0$ and
$\mathfrak{der}\left(\mathbb{O}\right)\cong\mathfrak{g}_{2}$ respectively,
then the Lie algebra of the group of elliptic motion
$\mathfrak{iso}\left(\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}P^{2}\right)$
is given by the direct sum

$\mathfrak{iso}\left(\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}P^{2}\right)\cong\mathfrak{sa}_{3}\left(\mathbb{C}\otimes\mathbb{O}\right)\oplus\mathfrak{g}_{2},$
(5.6)

where $\mathfrak{sa}_{3}\left(\mathbb{\mathbb{C}\otimes\mathbb{O}}\right)$ are
the anti-hermitian traceless three by three matrices in the bioctonionic
algebra444Brackets on this algebra are not relevant for our argument but can
be derived e.g. from [Vin, BS03]., i.e. $a_{ij}=-\overline{a}_{ji}^{\ast}$ and
$a_{00}+a_{11}+a_{22}=0$. Proceeding with the counting on the generators of
the algebra we obtain

$\text{dim}_{\mathbb{R}}\left(\mathfrak{iso}\left(\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}P^{2}\right)\right)=16\times
3+8\times 2+14=78,$ (5.7)

that gives the well known link between
$\mathfrak{\mathfrak{iso}}\left(\mathbb{\left(\mathbb{C}\otimes\mathbb{O}\right)}P^{2}\right)$
and the exceptional Lie group $E_{6}$. In the next section we will define all
generalised bioctonionic projective and hyperbolic planes from their isometry
group given as real forms of $E_{6}$ (see Tab. 2).

Plane | Isometry group
---|---
$\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$ | $E_{6\left(-78\right)}$
$\left(\mathbb{C}_{s}\otimes\mathbb{O}_{s}\right)P^{2}$ | $E_{6\left(6\right)}$
$\left(\mathbb{C}\otimes\mathbb{O}_{s}\right)P^{2}$ | $E_{6\left(2\right)}$
$\left(\mathbb{C}_{s}\otimes\mathbb{O}\right)P^{2}$ | $E_{6\left(-26\right)}$
$\left(\mathbb{C}\otimes\mathbb{O}\right)H^{2}$ | $E_{6\left(-14\right)}$
Table 2: All real forms of $E_{6}$ arise as isometries of generalised
bioctonionic projective and hyperbolic planes.

##  6 Bioctonionic Planes as Symmetric Spaces

Figure 3: Satake diagrams of real forms of $F_{4}$ $E_{6}$, their character
$\chi$ and corresponding projective plane of which they are the isometry
group.

All real forms of rank-3 Magic Squares have been classified and analyzed e.g.
in [CCM] (see also Refs. therein). Moreover, in [ABDN] the $D=3$ layer of the
‘magic pyramid’ of supergravities, containing various isometry Lie algebras of
some Rosenfeld projective planes, is identified with the $4\times 4$
Lorentzian rank-3 Magic Square
$\mathfrak{M}_{2,1}\left(\mathbb{A},\mathbb{B}\right)$, where $\mathbb{A}$ and
$\mathbb{B}$ are the four normed division Hurwitz’s algebras [Hur]. It is here
worth remarking that in [BM] a $6\times 6$ extension of the Magic Square was
also discussed, by introducing null extensions of quaternions and complex
numbers, respectively given by sextonions and tritonions (see also [BD]).

In the octonionic case, we start from the complexification of the Cayley plane

$\mathbb{O}P^{2}\left(\mathbb{C}\right)\simeq\frac{F_{4}(\mathbb{C})}{Spin_{9}(\mathbb{C})},$
(6.1)

and define four different real forms of the plane: one totally compact of type
$(0,16)$ and character $\chi=-16$ identified as $\mathbb{O}P^{2}$ and that is
known as the classical Cayley plane or as the octonionic projective plane; one
totally non-compact of type $\left(16,0\right)$ and character $\chi=16$
identified as $\mathbb{O}H^{2}$ and known as the hyperbolic octonionic plane;
and two of type $\left(8,8\right)$ and character $\chi=0$ named
$\mathbb{O}\widetilde{H}^{2}$ and $\mathbb{O}_{s}\widetilde{H}^{2}$. In all
cases the type identifies the signature, namely the cardinality of non-compact
and compact generators, i.e. $\left(\\#_{nc},\\#_{c}\right)$, and the
character $\chi$ is given by the difference between the two, i.e.
$\chi=\\#_{nc}-\\#_{c}$. The four plane are then defined as

$\displaystyle\mathbb{O}P^{2}$
$\displaystyle\simeq\frac{F_{4(-52)}}{Spin_{9}},$ (6.2)
$\displaystyle\mathbb{O}H^{2}$
$\displaystyle\simeq\frac{F_{4(-20)}}{Spin_{9}},$ (6.3)
$\displaystyle\mathbb{O}\widetilde{H}^{2}$
$\displaystyle\simeq\frac{F_{4(-20)}}{Spin_{8,1}},$ (6.4)
$\displaystyle\mathbb{O}_{s}\widetilde{H}^{2}$
$\displaystyle\simeq\mathbb{O}_{s}P^{2}\simeq\mathbb{O}_{s}H^{2}\simeq\frac{F_{4(4)}}{Spin_{5,4}}.$
(6.5)

For the bioctonionic case, things are a little more involved and starting from
the complex form of the bioctonionic Rosenfeld plane555The two semispinors
$\mathbf{16}_{\mathbb{C}}$ of $Spin(10)_{\mathbb{C}}$ in the tangent space of
(6.6) are an example of Jordan pair which is not made by a pair of Jordan
algebras (see e.g. [McCr])., i.e.666(6.6) has a Kähler structure pertaining to
the $\mathbb{C}$ factor in the isotropy/holonomy group.

$\left(\mathbb{C}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{2}\simeq\frac{E_{6}(\mathbb{C})}{Spin_{10}(\mathbb{C})\otimes\mathbb{C}},$
(6.6)

we have eight real different forms: one totally compact of type $(0,32)$ and
character $\chi=-32$ identified as
$\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$ and that we define as the
bioctonionic Rosenfeld projective plane; one totally non-compact of type
$(32,0)$ and character $\chi=32$ identified as
$\left(\mathbb{C}\otimes\mathbb{O}\right)H^{2}$ and that we define as the
bioctonionic Rosenfeld hyperbolic plane; four plane of type $(16,16)$ and
character $\chi=0$ and that are
$\left(\mathbb{C}\otimes\mathbb{O}\right)\widetilde{H}^{2}$,
$\left(\mathbb{C}\otimes\mathbb{O}_{s}\right)\widetilde{H}^{2}$,
$\left(\mathbb{C}_{s}\otimes\mathbb{O}\right)\widetilde{H}^{2}$ and
$\left(\mathbb{C}_{s}\otimes\mathbb{O}_{s}\right)\widetilde{H}^{2}$. The
various real forms of
$\left(\mathbb{C}\otimes\mathbb{O}\right)P_{\mathbb{C}}^{2}$ list as follows
[Ros93]:

$\displaystyle\left(\mathbb{C}\otimes\mathbb{O}\right)P^{2}$
$\displaystyle\simeq\frac{E_{6\left(-78\right)}}{Spin_{10}\otimes U_{1}},$
(6.7) $\displaystyle\left(\mathbb{C}\otimes\mathbb{O}\right)H^{2}$
$\displaystyle\simeq\frac{E_{6\left(-14\right)}}{Spin_{10}\otimes U_{1}},$
(6.8) $\displaystyle\left(\mathbb{C}\otimes\mathbb{O}\right)\widetilde{H}^{2}$
$\displaystyle\simeq\frac{E_{6\left(-14\right)}}{Spin_{8,2}\otimes U_{1}},$
(6.9) $\displaystyle\left(\mathbb{C}\otimes\mathbb{O}_{s}\right)P^{2}$
$\displaystyle\simeq\left(\mathbb{C}\otimes\mathbb{O}_{s}\right)H^{2}\simeq\left(\mathbb{C}\otimes\mathbb{O}_{s}\right)\widetilde{H}^{2}\simeq\frac{E_{6\left(2\right)}}{Spin_{6,4}\otimes
U_{1}},$ (6.10)
$\displaystyle\left(\mathbb{C}_{s}\otimes\mathbb{O}\right)P^{2}$
$\displaystyle\simeq\left(\mathbb{C}_{s}\otimes\mathbb{O}\right)H^{2}\simeq\left(\mathbb{C}_{s}\otimes\mathbb{O}\right)\widetilde{H}^{2}\simeq\frac{E_{6\left(-26\right)}}{Spin_{5,5}\otimes
SO_{1,1}},$ (6.11)
$\displaystyle\left(\mathbb{C}_{s}\otimes\mathbb{O}_{s}\right)P^{2}$
$\displaystyle\simeq\left(\mathbb{C}_{s}\otimes\mathbb{O}_{s}\right)H^{2}\simeq\left(\mathbb{C}_{s}\otimes\mathbb{O}_{s}\right)\widetilde{H}^{2}\simeq\frac{E_{6\left(6\right)}}{Spin_{5,5}\otimes
SO_{1,1}}.$ (6.12)

Spaces with $SO_{1,1}$ ($U_{1}$) factor in the stabilizer are pseudo-Kähler
(Kähler). It is here worth noticing that the Riemannian space
$\left(\mathbb{C}\otimes\mathbb{O}\right)H^{2}$ (6.8) appears as enlarged
scalar manifold (after 1-form dualization) of $\mathcal{N}=10$, $D=2+1$ “pure”
supergravity (see e.g. [dWTN]), as well as the non-BPS $Z_{H}=0$ “moduli
space” of extremal black hole attractors in $\mathcal{N}=2$, $D=3+1$
exceptional magic theory [FM]. From the theory of Jordan triple systems, such
a manifold is related to a pair of octonionic vectors (see [GST2] and Refs.
therein).

Finally, there are other two pseudo-Riemannian real forms of the bioctonionic
plane (6.6), of type $\left(20,12\right)$ and $\left(12,20\right)$, both
Kähler, respectively with character $\chi=8$ and $\chi=-8$, namely :

$\displaystyle\frac{E_{6(2)}}{SO_{10}^{\ast}\otimes U_{1}},$ (6.13)
$\displaystyle\frac{E_{6(-14)}}{SO_{10}^{\ast}\otimes U_{1}},$ (6.14)

that apparently do not have a projective or hyperbolic equivalent on
bioctonionic algebras. This fact can be traced back to the absence of the Lie
algebra $\mathfrak{so}_{4n+2}^{\ast}$ in the entries of the real forms of the
Magic Square (cfr. [CCM], and Refs. therein), and we leave this intriguing
issue for further future work. Here, we only notice that (6.14) appears as the
enlarged scalar manifold of $\mathcal{N}=5$, $D=3+1$ “pure” supergravity
timelikely reduced to $\mathcal{N}=10$, $D=3+0$ “pure” supergravity (after
complete dualization of 1-forms to 0-forms); cf. [BGM].

##  7 Musings on the Physics of $E_{6(-78)}$ and $E_{6(2)}$

The so-called “exceptional sequence” is given by the Lie algebras
$\mathfrak{e}_{n}$ for $n=3,\dots,8$, which respectively correspond to
$\mathfrak{sl}_{3}\oplus\mathfrak{sl}_{2}$, $\mathfrak{sl}_{5}$,
$\mathfrak{so}_{10}$, $\mathfrak{e}_{6}$, $\mathfrak{e}_{7}$, and
$\mathfrak{e}_{8}$. The application of exceptional Lie algebras in physics was
pioneered by Gürsey. $SU_{5}$ Grand Unified Theories (GUT) unifies the bosons
into a single representation and $Spin_{10}$ GUT unifies one generation of the
fermions, which are contained within $E_{6}$ GUT [GRS]. Bars and Günaydin
explored $E_{8}$ GUT for three generations of the Standard Model [BG]. While
it is commonly thought that $E_{6}$ is the only exceptional GUT algebra with
complex representations [Wit], Barr investigated the role of $E_{8}\rightarrow
E_{6}\otimes SU_{3}$, showing how the $SU_{3}$ flavor symmetry leads to three
generations with mirror fermions and $E_{6}$ GUT [Barr].

More recently, Dubois-Violette and Torodov explored the state space of three
generations of fermions via $\mathfrak{J}_{3}(\mathbb{O})$ in relation to
$F_{4}$ [TDV]. Boyle elaborated on the role of $E_{6}$ via states from
$\mathfrak{J}_{3}^{\mathbb{C}}(\mathbb{O})$ independent of $E_{6}$ GUT [BF,
Bo]. Krasnov has also discussed the role of $\mathbb{O}_{s}\otimes\mathbb{O}$
and $Spin_{11,3}$, but did not obtain three generations [Krb]. Two of the
authors have previously discussed the role of $E_{8(-24)}$ with $Spin_{12,4}$
and a spinor from $(\mathbb{O}_{s}\otimes\mathbb{O})\mathbb{P}^{2}$ for three
generations of matter [CMR].

Wilczek et al. articulate how $Spin_{3}$ flavor symmetry is preferred over
$SU_{3}$ for anomaly cancellation without mirror fermions [RVW]. $E_{8(-24)}$
contains $Spin_{4,4}\supset Spin_{4,1}\otimes Spin(3)$, implying extra time
dimensions relate to mass eigenstates, as energy/mass are the time components
of energy-momentum in phase space [Ko, CMR]. Wilson found a similar
interpretation with $Spin_{3,3}$ and geometric algebra [Wil]. Moreover, since
$Spin_{6,2}\cong SU_{2,2}(\mathbb{H})_{\mathbb{R}}$ (see e.g. [Var]), one can
reasonably guess that $Spin_{4,4}\cong SU_{2,2}(\mathbb{H}_{s})_{\mathbb{R}}$.
Thus, it holds that

$\begin{array}[]{ccc}Spin_{3,1}\cong
SL_{2}(\mathbb{C})_{\mathbb{R}}&\leftarrow&Spin_{3,3}\cong
SL_{2}(\mathbb{H}_{s})_{\mathbb{R}}\cong SL_{4}(\mathbb{R})\\\
\uparrow&&\uparrow\\\ Spin_{4,2}\cong SU_{2,2}&\leftarrow&Spin_{4,4}\cong
SU_{2,2}(\mathbb{H}_{s})_{\mathbb{R}}\cong
Spin(\mathbb{O}_{s})_{\mathbb{R}}\end{array}$ (7.1)

When singling $SL_{2}(\mathbb{C})_{\mathbb{R}}$ out from
$SL_{2}(\mathbb{H}_{s})_{\mathbb{R}}$, half of the spinorial degrees of
freedom are lost, as $\mathbb{H}_{s}$ is made à la Cayley-Dickson from two
$\mathbb{C}_{s}$ and only contains one $\mathbb{C}$. The triality of
$Spin_{4,4}$ leads to three charts of $Spin_{4,2}\otimes U_{1}$, allowing for
three generations for the price of two in a manner that avoids mirror fermions
due to only containing a single complex representation in $\mathbb{H}_{s}$ via
$SL_{2}(\mathbb{C})_{\mathbb{R}}\subset SL_{2}(\mathbb{H}_{s})_{\mathbb{R}}$
or $SU_{2,2}\subset SU_{2,2}(\mathbb{H}_{s})_{\mathbb{R}}$. A unification of
classical phase space with spacetime and energy-momentum for one generation
can be found in $Spin_{6,2}$ or $Spin_{4,4}$. Therefore, three mass
generations leads to two more energy dimensions, resulting in $Spin_{6,4}$, a
subalgebra of $E_{6(2)}$. The Peirce decomposition of $\mathbf{27}$ w.r.t.
$E_{6(2)}$ is used to identify only $\mathbf{16}$ representations as
fermionic, while $E_{6(2)}$ itself also contains spinors; in this framework,
$Spin_{4,4}$ triality can be proposed to describe three generations of matter.

Recent works discussing $\mathfrak{J}_{3}(\mathbb{O})$ and
$\mathfrak{J}_{3}^{\mathbb{C}}(\mathbb{O})$ motivate three generations within
a single $\mathbf{26}$ or $\mathbf{27}$ representation of $F_{4}$ or $E_{6}$
[TDV, Bo]. While three charts of $Spin_{9}$ and $Spin_{10}\otimes U_{1}$ are
in $F_{4}$ and $E_{6}$, an appropriate counting of on-shell and off-shell
states must be found. For instance, a complex spinor $\mathbf{16}$ of
$Spin_{10}$ is contained within the Peirce decomposition of $\mathbf{27}$ of
$E_{6}$. However, three sets of $\mathbf{16}$ as two-component spinors overlap
significantly within $\mathbf{27}$. In other words, we suggest that the
triality of $Spin_{4,4}$, rather than $Spin_{8}$, gives three generations.
$\mathfrak{J}_{3}^{\mathbb{C}}(\mathbb{O})$ cannot fully contain the field
content of three generations of spinors, but Weyl spinors of
$Spin_{4,4}\otimes Spin_{8}$ can, which stems from a single Majorana-Weyl
spinor of $Spin_{12,4}$ within $E_{8(-24)}$ [CMR].

The three generations of Standard Model fermions within
$(\mathbb{O}_{s}\otimes\mathbb{O})\mathbb{P}^{2}$ allows for three
$(\mathbb{C}\otimes\mathbb{O})\mathbb{P}^{2}$ spinors, since there are three
complex units in $\mathbb{O}_{s}$. A single
$(\mathbb{C}\otimes\mathbb{O})\mathbb{P}^{2}$ cannot encode three generations,
as three combinations of $\mathbf{16}$ are needed within the theory. If
$\mathbf{27}$ contains a mix of fermions and bosons, rather than $\mathbf{27}$
fermions as found in $E_{6}$ GUT, the $\mathbf{27}$ of $E_{6(-78)}$, then the
$\mathbf{27}$ as weight vectors does not have the fermionic roots determined.
Once one of three $Spin_{10}\otimes U_{1}$ is chosen, then, this determines
which roots are fermions. A key departure from $E_{6}$ GUT is the
interpretation of the $\mathbf{27}$ representation as purely fermionic; since
the fermions of the standard model are assigned to the pseudo-Riemannian space
$E_{8(-24)}/Spin_{12,4}$, the Peirce decomposition of
$\mathbf{27}=\mathbf{16}\oplus\mathbf{10}\oplus\mathbf{1}$ identifies only the
$\mathbf{16}$ as containing fermions. On the other hand, since $Spin_{10}$ is
the largest GUT that contains no additional fermions, the role of $E_{6(-78)}$
is primarily suggested to encode three charts of flavor eigenstates of
$Spin_{10}\otimes U_{1}$. In this manner, the utility of $E_{6(-78)}$ is
similar if not identical to the one suggested by Boyle [Bo]. Our distinction
is that only a single generation of fermions comes from a single
$\mathbf{16}\in\mathbf{27}$ of $Spin_{10}\subset E_{6(-78)}$; instead, three
generations of leptons are found within777Private correspondence with Todorov
confirms that the $\mathbf{27}$ does not contain three generations, but
provides the manifold for three generations.
$\mathfrak{e}_{6(2)}\oplus\mathbf{27}\oplus\overline{\mathbf{27}}$. Since
$E_{6(2)}\otimes SU_{3}$ contains the $SU_{3,c}$ of the strong force with
color charge, $Spin_{6,4}\otimes U_{1}\subset E_{6(2)}$ can be identified as a
type of “gravi-weak” symmetry [NP, Da, Al]. On the other hand, the electroweak
symmetry $SU_{2}\otimes U_{1}$ is the only subsector of $Spin_{10}$ that is a
chiral gauge theory at low energies, which stems back to spinors of spacetime.
Thus, by putting color aside, $E_{6(2)}$ can be studied to encode mass
eigenstates, which provides a new physical motivation for the non-compact real
form(s) of $E_{6}$ and split octonions. Various paths lead from $E_{8(-24)}$
to $SU_{2,2}\otimes U_{1}\otimes SU_{3}\otimes U_{1}$ at low energies, many of
which pass through $SU_{2,2}\otimes U_{1}\otimes SU_{3}\otimes SU_{2}\otimes
U_{1}$ [CMR].

##  8 Conclusion

Making use of the Veronese coordinates, we have explicitly constructed two
different bioctonionic planes, namely the complexification of the Cayley plane
and the bioctonionic Rosenfeld plane, showing some of their different
geometrical features, which yield to different algebraic structures on the
three by three bioctonionic Hermitian matrices. We have also discussed the
isometry groups of the two planes, and then characterized systematically all
possible octonionic and bioctonionic planes as symmetric spaces (of Kähler or
pseudo-Kähler type in the bioctonionic case). This approach brought us to
single out two pseudo-Riemannian real forms of the bioctonionic plane that
apparently do not have a projective or hyperbolic equivalent on the
bioctonionic algebra. One of these spaces appears as enlarged scalar manifold
of a “pure” (i.e. not matter-coupled) supergravity theory with 20
supersymmetries when dimensionally reduced from $3+1$ to $3+0$ space-time
dimensions (namely, when reduced along time); we plan to investigate more on
this in the future. Finally, we have briefly commented on the physical
relevance of some bioctonionic Rosenfeld plane, hinting to some interesting
applications of real forms of $E_{6}$.

## Acknowledgments

The work of D.Corradetti is supported by a grant of the Quantum Gravity
Research Institute. The work of AM is supported by a “Maria Zambrano”
distinguished researcher fellowship, financed by the European Union within the
NextGenerationEU program.

## References

  * [Al] Alexander S., Marcianò A., Smolin L., Gravitational origin of the weak interaction’s chirality. Phys. Rev. D89 (2014) 065017.
  * [ABDN] Anastasiou A., Borsten L., Duff M.J., Hughes L.J., Nagy S., A magic pyramid of supergravities, JHEP 04 (2014) 178, arXiv:1312.6523 [hep-th].
  * [Bae] Baez J.C., The Octonions, Bull. Am. Math. Soc. 39 (2002) 145-205, Bull. Am. Math. Soc. 42 (2005) 213 (erratum), math/0105155 [math.RA].
  * [Barr] Barr S. M., $\mathit{E}_{8}$ family unification, mirror fermions, and new low-energy physics. Phys. Rev. D37 (1988) 204.
  * [BD] Bentz L., Dray T., Subalgebras of the Split Octonions, Adv. Appl. Clifford Algebras 28 (2018) 2, 40.
  * [BF] Boyle L., Farnsworth S., The standard model, the Pati-Salam model, and “Jordan geometry”, New J. Phys. 21 (2020) 073023, arXiv: 1910.11888 [hep-th].
  * [BG] Bars I., Günaydin M., Grand Unification with the Exceptional Group $\mathit{E}_{8}$. Phys. Rev. Lett. 45 (1980) 859.
  * [BGM] Breitenlohner P., Maison D., Gibbons G.W., Four-Dimensional Black Holes from Kaluza-Klein Theories, Commun. Math. Phys. 120 (1988) 295.
  * [BH] Baez J., Huerta J., Division Algebras and Supersymmetry I, Proc. Symp. Pure Math. 81, AMS, Providence, RI (2010), 65-80, arXiv:0909.0551 [hep-th].
  * [BM] Borsten L., Marrani A., A Kind of Magic, Class. Quant. Grav. 34 (2017) 23, 235014, arXiv:1707.02072 [hep-th].
  * [Bo] Boyle L., The Standard Model, The Exceptional Jordan Algebra, and Triality, arXiv:2006.16265 [hep-th].
  * [BS03] Barton C.H., Sudbery A. Magic squares and matrix models of Lie algebras, Adv. Math. 180 (2) (2003) 596-647, math/0203010 [math.RA].
  * [Ca19a] Castro Perelman C., $\mathbb{R}\otimes\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$-valued gravity as a grand unified field theory, Adv. Appl. Clifford Algebras 29 (2019) no.1, 22\.
  * [Ca19b] Castro Perelman C., On $\mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$-valued gravity, sedenions, hermitian matrix geometry and nonsymmetric Kaluza- Klein theory, Adv. Appl. Clifford Algebras 29 (2019) no.3, 58.
  * [Cay] Cayley A., On Jacobi’s Elliptic functions, in reply to the Rev. Brice Bronwin; and on Quaternions, Philosophical Magazine 26 (1845) 208.
  * [CCM] Cacciatori S. L., Cerchiai B. L., Marrani A., Squaring the Magic, Adv. Theor. Math. Phys. 19 (2015) 923-954, arXiv:1208.6153 [math-ph].
  * [Cha] Chaput, P.-E., Geometry over composition algebras: Projective geometry, J. of Algebra 298 (2), 340-362 (2006), math/0509549[math.AG].
  * [CMR] Chester, D., Marrani, A., Rios, M., Beyond the standard model with six-dimensional spacetime, arXiv:2002.02391v2 [phys.gen-ph].
  * [Co] Corradetti D., Complexification of the Exceptional Jordan Algebra and its Application to Particle Physics, J. Geom. Symmetry Phys. 61, 1-16 (2021).
  * [Da] Das C. R., Laperashvili L. V., Tureanu A., Graviweak unification, invisible universe, and dark energy. Int. J. Mod Phys. A28 (2013) No. 18, 1350085, arXiv: 1304.3069 [hep-th].
  * [Dix] Dixon, G.M.: “Division algebras: Octonions, quaternions, complex numbers and the algebraic design of physics”, Springer, Boston, MA (1994).
  * [DW] Wangberg A., Dray T., $E_{6}$, the group: The structure of $SL(3,\mathbb{O})$, J. Alg. and Its Appl. 14(06), 1550091 (2015), arXiv:1212.3182 [math.RA].
  * [dWTN] de Wit B., Tollsten A.K., Nicolai H., Locally supersymmetric $\mathit{D=3}$ nonlinear sigma models, Nucl. Phys. B392 (1993) 3, hep-th/9208074.
  * [FM] Ferrara S., Marrani A., On the Moduli Space of non-BPS Attractors for $\mathcal{N}\mathit{=2}$ Symmetric Manifolds, Phys. Lett. B652 (2007) 111, arXiv: 0706.1667 [hep-th].
  * [FPT] Franek P., Pazourek K., Tuček V., Hyperplane section $\mathbb{O}P^{2}$ of the complex Cayley plane as the homogeneous space $F_{4}/P_{4}$, Commentat. Math. Univ. Carol. 52, No. 4, 535-549 (2011), arXiv:1006.3407 [math.AG].
  * [Fre54] Freudenthal H., Beziehungen der E7 und E8 zur Oktavenebene, Indagat. Math. 16 (1954) 218-230 and 636-638, 17 (1955) 151-157 and 277-285, 25 (1963) 457-487.
  * [Fu14] Furey C., Generations: three prints, in colour, JHEP 10, 046 (2014), arXiv:1405.4601 [hep-th].
  * [Fu16] Furey C., “Standard model physics from an algebra?”, Ph.D. thesis, University of Waterloo, 2015, arXiv:1611.09182 [hep-th].
  * [GKLY] Günaydin M., Kallosh R., Linde A., Yamada Y., M-theory Cosmology, Octonions, Error Correcting Codes, JHEP 01 (2021) 160, arXiv:2008.01494 [hep-th].
  * [Gra] Hamilton W.R., Note, by Sir W. R. Hamilton, respecting the researches of John T. Graves, Esq., Transactions of the Royal Irish Academy 21 (1948) 338.
  * [Gre19] Gillard A.B., Gresnigt N.G., Three fermion generations with two unbroken gauge symmetries from the complex sedenions, Eur. Phys. J. C79 (5), 446 (2019), arXiv:1904.03186 [hep-th].
  * [Gre20] Gresnigt N.G., Sedenions, the Clifford algebra Cl(8), and three fermion generations, PoS EPS-HEP 2019 (2020) 615.
  * [GRS] Gürsey F., Ramond, P. Sikivie P., A Universal Gauge Theory Model Based on E6. Phys. Lett. B 60 (1976).
  * [GST] Günaydin M., Sierra G., Townsend P.K., Exceptional Supergravity Theories and the Magic Square, Phys. Lett. B133, 72-76 (1983).
  * [GST2] Günaydin M., Sierra G., Townsend P.K., The Geometry of $\mathcal{N}\mathit{=2}$ Maxwell-Einstein Supergravity and Jordan Algebras, Nucl. Phys. B242 (1984) 244.
  * [Hur] Hurwitz A., Uber die Composition der quadratischen Formen von beliebig vielen Variabeln, Nachr. Ges. Wiss. Gottingen, 309-316 (1898).
  * [Jac68] Jacobson N.: “Structure and Representations of Jordan Algebras”, American Mathematical Society Providence, RI (1968).
  * [Jac61] Jacobson N., Some groups of transformations defined by Jordan algebras. III, J. Reine Angew. Math. 207, 61-85 (1961).
  * [JvNW] Jordan P., von Neumann J., Wigner E., On an algebraic generalisation of the quantum mechanical formalism, Ann. Math. 35, 29–64 (1934).
  * [Ko] Kostant B., The Principle of Triality and A Distinguished Unitary Representation of SO(4,4), contribution to : “Differential geometrical methods in theoretical physics” (Como, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 250, Kluwer Acad. Publ., Dordrecht (1988), pp. 65–108.
  * [Kra] Krasnov K., $\mathit{SO(9)}$ characterisation of the Standard Model gauge group, J. Math. Phys. 62, 021703 (2021), arXiv:2104.01786 [hep-th].
  * [Krb] Krasnov, K. Spin(11,3), particles and octonions. arXiv:2104.01786 [hep-th].
  * [Krut] Krutelevich S., Jordan algebras, exceptional groups, and higher composition laws, J. Algebra 314, 924 (2007), math/0411104[math.NT].
  * [Ma] Masi N., An exceptional $\mathit{G(2)}$ extension of the Standard Model from the correspondence with Cayley–Dickson algebras automorphism groups, Sci. Rep. 11, 22528 (2021), arXiv:2111.11849[hep-ph].
  * [McCr] McCrimmon K.: “A taste of Jordan algebras”, Universitext. Springer-Verlag, New York (2004).
  * [NP] Nesti, F., Percacci, R. 2008. Gravi-weak unification J. Phys. A: Math. Theor. 41 075405.
  * [RVW] Reig M., Valle J.W.F., and Wilczek F., SO(3) family symmetry and axions, Phys. Rev. D98, no.9, 095008 (2018) arXiv:1805.08048 [hep-ph].
  * [Ros93] Rosenfeld B. A., Spaces with Exceptional Fundamental Groups, Publications de l’Institut Mathématique, nouvelle série tome 54 (68), 97-119 (1993).
  * [Ros97] Rosenfeld B.A., “Geometry of Lie Groups”, Springer-Verlag US (1997).
  * [Ros98] Rosenfeld B.A., Geometry of Planes over Nonassociative Algebras, Acta Applicand Mathematic 50, 103-110 (1998).
  * [SBG11] Salzmann H., Betten D., Grundhöfer T., Hähl H., Löwen R., Stroppel M.: “Compact Projective Planes: With an Introduction to Octonion Geometry”, De Gruyter, Berlin - New York (2011).
  * [SV] Springer T.A., Veldkamp FD.: “Octonions, Jordan Algebras and Exceptional Groups”, Springer-Verlag, Berlin - Heidelberg (2000).
  * [TDV] Todorov I.T., Dubois-Violette M., Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra, Int. J. Mod. Phys. A33, 1850118 (2018), arXiv:1806.09450[hep-th].
  * [Tit53] Tits J., Le plan projectif des octaves et les groupes de Lie exceptionnels, Acad. Roy. Belgique. Bull. Cl. Sci. 39(5), 309-329 (1953).
  * [Var] Varadarajan, V. S.: “Supersymmetry for mathematicians : an introduction”, American Mathematical Society, Providence RI (2004).
  * [Vin] Vinberg E.B., A Construction of Exceptional Lie Groups (Russian), Tr. Semin. Vek- torn. Tensorn. Anal., 13:7–9, 1966.
  * [Wil] Wilson R. A., On the Problem of Choosing Subgroups of Clifford Algebras for Applications in Fundamental Physics, Adv. Appl. Clifford Algebras 31 59 (2021), arXiv: 2011.05171 [math.RA].
  * [Wit] Witten E., Quest for unification, contribution to: “10th International Conference on Supersymmetry and Unification of Fundamental Interactions (SUSY02)”, 604-610, arXiv:0207124 [hep-ph].
  * [Ya] Yale P.B., Automorphisms of the complex numbers, Math. Mag. 39, 135–141 (1966).
  * [Yok] Yokota I., Exceptional Lie Groups, arXiv:0902.0431 [math.DG].


# Duality in $BP\langle n\rangle$ (co)homology

Donald M. Davis  Department of Mathematics, Lehigh University
Bethlehem, PA 18015, USA<EMAIL_ADDRESS>
(Date: May 23, 2022)

###### Abstract.

Let $E=BP\langle n\rangle$ denote the Johnson-Wilson spectrum, localized at
$p$. It is proved that if $E_{*}(X)$ is locally finite, then there is an
isomorphism of right $E_{*}$-modules
$E^{*}(X)\approx(E_{*}(\Sigma^{D+n+1}X))^{\vee}$, where $D=\sum|v_{i}|$ and
$M^{\vee}=\operatorname{Hom}(M,{\mathbb{Q}}/{\mathbb{Z}})$ is the Pontryagin
dual. This result was motivated by work of the author and W.S.Wilson regarding
the 2-local $ku$-homology and -cohomology groups of the Eilenberg-MacLane
space $K({\mathbb{Z}}/2,2)$.

###### Key words and phrases:

Universal Coefficient Theorem, Brown-Peterson(co)homology, Ext groups, K
theory

2000 Mathematics Subject Classification: 55U20, 55U30, 55N20, 18G15.

##  1\. Main results

Let $E=BP\langle n\rangle$ denote the Johnson-Wilson spectrum ([5]) localized
at a prime $p$, which satisfies that
$E_{*}=\pi_{*}(E)={\mathbb{Z}}_{(p)}[v_{1},\ldots,v_{n}]$, with
$|v_{i}|=2(p^{i}-1)$. Our motivating example is the case $p=2$, $n=1$, when
$E$ is the spectrum $ku$ for connective complex $K$-theory, localized at 2.
Our main result is an isomorphism between certain $E$-cohomology groups and
the Pontryagin dual of $E$-homology groups. We require that $E_{*}(X)$ is
locally finite, which means that for each $i$, the $E_{*}$-module generated by
$E_{i}(X)$ is finite. If $M$ is an $R$-module, we denote by $M^{\vee}$ the
right $R$-module $\operatorname{Hom}(M,{\mathbb{Q}}/{\mathbb{Z}})$. Localized
at $p$, we prefer to write ${\mathbb{Q}}/{\mathbb{Z}}$ as
${\mathbb{Z}}/p^{\infty}$.

######  Theorem 1.1.

If $E=BP\langle n\rangle$ and $E_{*}(X)$ is locally finite, there is an
isomorphism of right $E_{*}$-modules

$E^{*}(X)\approx(E_{*}(\Sigma^{D+n+1}X))^{\vee},$

where $D=\sum|v_{i}|=2((p^{n+1}-1)/(p-1)-(n+1))$.

We prove this result using a Universal Coefficient Theorem and the following
algebraic result, which we prove in Section 2. If $M$ is a graded module,
$\Sigma^{D}M$ denotes the graded module obtained from $M$ by increasing
gradings by $D$. Our Ext groups are in the category of graded modules, and the
second superscript refers to the grading.

######  Theorem 1.2.

Let $R={\mathbb{Z}}_{(p)}[x_{1},\ldots,x_{n}]$ with $|x_{i}|$ positive
integers, and let $D=\sum|x_{i}|$. If $M$ is a locally finite graded
$R$-module, there is an isomorphism of graded right $R$-modules

$\operatorname{Ext}_{R}^{s}(M,R)\approx\begin{cases}\Sigma^{D}M^{\vee}&s=n+1\\\
0&s\neq n+1.\end{cases}$

###### Proof of Theorem 1.1.

By [7, Corollary, p.257], if $E$ is an $A_{\infty}$ ring spectrum, there is a
Universal Coefficient spectral sequence

$\operatorname{Ext}_{E_{*}}^{s,t}(E_{*}X,E_{*})\Rightarrow E^{s+t}X.$

By [1, Corollary 3.5], $BP\langle n\rangle$ is an $A_{\infty}$ ring spectrum.
By Theorem 1.2 with $R=E_{*}$, the spectral sequence must collapse, as it is
confined to a single value of $s$, and the $E_{\infty}$ groups are as claimed.

In Section 3, we illustrate Theorem 1.1 for a portion of $ku_{*}(K_{2})$ with
$K_{2}=K({\mathbb{Z}}/2,2))$, localized at 2. Here we state the application of
Theorem 1.1 to this case as a corollary.

######  Corollary 1.3.

There is an isomorphism of right $ku_{*}$-modules
$ku^{*}(K_{2})\approx(ku_{*}(\Sigma^{4}K_{2}))^{\vee}$.

Observe also that the case $n=0$ of Theorem 1.1 is the usual Universal
Coefficient Theorem when $H_{*}(X;{\mathbb{Z}}_{(p)})$ is finite.

After a version of this paper was placed on the arXiv, John Greenlees pointed
out to the author that Theorem 1.1 could apparently be deduced using concepts
of duality in stable homotopy theory, and quickly prepared a short manuscript
([4]) which did so, at least when $n\leq 2$. A result using Brown-Comenetz
duality ([6, Corollary 9.3]) is closely related. We feel that the elementary
nature of our presentation lends worth to our paper.

The author is grateful to Andy Baker, John Greenlees, Andrey Lazarev, Doug
Ravenel, Chuck Weibel, and Steve Wilson for helpful suggestions.

##  2\. Proof of Theorem 1.2

The following result is certainly well-known. Here ${\mathbb{Z}}_{p}$ denotes
the $p$-adic integers.

######  Proposition 2.1.

Let $R={\mathbb{Z}}_{(p)}[x_{1},\ldots,x_{n}]$ with $|x_{i}|$ positive
integers, and let $D=\sum|x_{i}|$. In the category of graded $R$-modules

$\operatorname{Ext}_{R}^{s,t}({\mathbb{Z}}/p,R)=\begin{cases}{\mathbb{Z}}/p&(s,t)=(n+1,D)\\\
0&\text{otherwise},\end{cases}$ (2.2)

and

$\operatorname{Ext}_{R}^{s}({\mathbb{Z}}/p^{\infty},R)=\begin{cases}{\mathbb{Z}}_{p}&(s,t)=(n+1,D)\\\
0&\text{otherwise}.\end{cases}$ (2.3)

###### Proof.

Let $\mathcal{C}_{0}$ be the chain complex $C_{1}\to C_{0}$ with $C_{1}$ and
$C_{0}$ free ${\mathbb{Z}}_{(p)}$-modules of rank 1 and grading 0 with
generators $g_{0}$ and $\iota_{0}$, respectively, and $d(g_{0})=p\iota_{0}$.
For $1\leq i\leq n$, let $\mathcal{C}_{i}$ be the chain complex $C_{i,1}\to
C_{i,0}$ with $C_{i,1}$ and $C_{i,0}$ free ${\mathbb{Z}}_{(p)}[x_{i}]$-modules
of rank 1 with generators $g_{i}$ and $\iota_{i}$, respectively, and
$d(g_{i})=x_{i}\iota_{i}$. Here $|\iota_{i}|=0$ and $|g_{i}|=|x_{i}|$. Then
${\mathbb{C}}:=\mathcal{C}_{0}\otimes\mathcal{C}_{1}\otimes\cdots\otimes\mathcal{C}_{n}$
is a chain complex of free $R$-modules with
$H_{j}({\mathbb{C}})={\mathbb{Z}}/p$ for $j=0$, and 0 for $j>0$, by the
Künneth Theorem. Thus ${\mathbb{C}}$ is an $R$-resolution of ${\mathbb{Z}}/p$.
Hence $\operatorname{Ext}_{R}^{s}({\mathbb{Z}}/p,R)$ is the $s^{\text{th}}$
cohomology group of the dual complex $\operatorname{Hom}_{R}({\mathbb{C}},R)$,
which is the tensor product,
$\mathcal{C}^{*}_{0}\otimes\mathcal{C}^{*}_{i}\otimes\cdots\otimes\mathcal{C}_{n}^{*}$,
of the dual complexes. The cohomology group is nonzero only when $s=n+1$,
where it is ${\mathbb{Z}}/p$, dual to $g_{0}\otimes g_{1}\otimes\cdots\otimes
g_{n}$, in grading $D$.

For the second result, we replace $\mathcal{C}_{0}$ by a chain complex
$\mathcal{C}^{\prime}$ which has $C_{1}^{\prime}$ and $C_{0}^{\prime}$ free
${\mathbb{Z}}_{(p)}$-modules with generators indexed by positive integers,
$g^{\prime}_{j}$ and $\iota^{\prime}_{j}$, respectively, with
$d(g^{\prime}_{j})=\iota^{\prime}_{j}-p\iota^{\prime}_{j+1}$. Then
$H_{0}(\mathcal{C}^{\prime})={\mathbb{Z}}/p^{\infty}$ is the nonzero homology
group, and $H^{1}(\mathcal{C}^{\prime})={\mathbb{Z}}_{p}$ is the nonzero
cohomology group. The rest of the proof follows as in the previous paragraph.

###### Proof of Theorem 1.2.

We first consider the case when $M$ is finite, and proceed by induction on the
size of $M$. The result is true when $M={\mathbb{Z}}/p$ by (2.2). Let $\alpha$
denote a generator of
$\operatorname{Ext}_{R}^{n+1,D}({\mathbb{Z}}/p^{\infty},R)$ from (2.3). Yoneda
product $\alpha\circ$ is a natural transformation of right $R$-modules

$\operatorname{Ext}_{R}^{*,*}(-,{\mathbb{Z}}/p^{\infty})\to\operatorname{Ext}_{R}^{*+n+1,*+D}(-,R).$

If

$0\to K\to M\to Q\to 0$

is a short exact sequence of finite $R$-modules, by induction we may assume
the theorem is true for $K$ and $Q$, and hence by the exact Ext sequence,
$\operatorname{Ext}_{R}^{s}(M,R)=0$ if $s\neq n+1$. We also obtain a
commutative diagram of short exact sequences of right $R$-modules

${0}$${\Sigma^{D}Q^{\vee}}$${\Sigma^{D}M^{\vee}}$${\Sigma^{D}K^{\vee}}$${0}$${0}$${\operatorname{Ext}_{R}^{n+1}(Q,R)}$${\operatorname{Ext}_{R}^{n+1}(M,R)}$${\operatorname{Ext}_{R}^{n+1}(K,R)}$${0.}$

The 0’s on the ends of the first sequence follow from [8, p.70], and for the
second sequence by the induction. By the 5-lemma, our result is true for
finite $R$-modules.

Now let $M$ be locally finite, and for any positive integer $k$, let $K_{k}$
(resp. $Q_{k}$) denote the set of all elements of $M$ in grading $>k$ (resp.
$\leq k$). There is a short exact sequence of $R$-modules

$0\to K_{k}\to M\to Q_{k}\to 0.$

Since $Q_{k}$ is finite, the induced $\operatorname{Ext}_{R}(-,R)$ sequence
implies that for $s\neq n+1$, $\operatorname{Ext}_{R}^{s,j}(M,R)=0$ for $j\leq
k$. Since $k$ was arbitrary, we deduce that
$\operatorname{Ext}_{R}^{s}(M,R)=0$ for $s\neq n+1$. Again Yoneda product with
$\alpha$ yields a commutative diagram of short exact sequences of right
$R$-modules

${0}$${\Sigma^{D}Q_{k}^{\vee}}$${\Sigma^{D}M^{\vee}}$${\Sigma^{D}K_{k}^{\vee}}$${0}$${0}$${\operatorname{Ext}_{R}^{n+1}(Q_{k},R)}$${\operatorname{Ext}_{R}^{n+1}(M,R)}$${\operatorname{Ext}_{R}^{n+1}(K_{k},R)}$${0.}$

The left vertical arrow is iso since $Q_{k}$ is finite, and the groups in the
right vertical arrow are 0 in grading $\leq k$. Since $k$ is arbitrary, we
deduce that the center vertical arrow is an isomorphism.

##  3\. An example when $n=1$, $p=2$, and $X=K({\mathbb{Z}}/2,2)$.

In [9] and [2], the author and, previously, Wilson initiated a partial
calculation of $ku_{*}(K_{2})$, where $K_{2}=K({\mathbb{Z}}/2,2)$, in their
studies of Stiefel-Whitney classes. In [3], these authors made a complete
calculation of $ku^{*}(K_{2})$. Using our new Theorem 1.1, we can now give a
complete determination of $ku_{*}(K_{2})$, since we know that it is locally
finite, as it was noted in [2] that it contains no infinite groups or infinite
$v_{1}$-towers.

The work in [2] and [3] was done using the Adams spectral sequence. It is
interesting to compare the forms of the two Adams spectral sequence
$E_{\infty}$ calculations. What appears as an $h_{0}$ multiplication in one
usually appears as an exotic extension (multiplication by 2 not seen in Ext)
in the other. We illustrate here with corresponding small portions of each.
The portion of $ku^{*}(K_{2})$ in Figure 3.1 is called $A_{5}$ in [3]. Note
that in our $ku^{*}$ chart, indices increase from right to left. Exotic
extensions appear in red. One should think of the dual of the $ku_{*}$ chart
as an upside-down version of the chart. The dual of the element in position
$(30,7)$ in Figure 3.2 is in position $(34,0)$ in Figure 3.2.

######  Figure 3.1.

A portion of $ku^{*}(K_{2})$

$36$$32$$28$$24$$20$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$

######  Figure 3.2.

Corresponding portion of $ku_{*}(K_{2})$

$16$$20$$24$$28$$32$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$$\scriptstyle\bullet$

## References

  * [1] A.Baker and A.Jeanneret, Brave new Hopf algebroids and extensions of $MU$-algebras, Homology, Homotopy, and Applications 4 (2002) 163–173.
  * [2] D.M.Davis and W.S.Wilson, Stiefel-Whitney classes and immersions of orientable and Spin manifolds, Topology and Appl 307 (2022) https://doi.org/10.1016/j.topol.2021.107780.
  * [3] , Connective $K$-theory of the Eilenberg-MacLane space $K({\mathbb{Z}}/2,2)$, in preparation, www.lehigh.edu/$\sim$dmd1/kkpaper3.pdf.
  * [4] J.P.C.Greenlees, Gorenstein duality and universal coefficient theorems, in preparation.
  * [5] D.C.Johnson and W.S.Wilson, Projective dimension and Brown-Peterson homology, Topology 12 (1973) 327–353.
  * [6] M.Mahowald and C.Rezk, Brown-Comenetz duality and the Adams spectral sequence, Amer Jour Math 121 (1999) 1153–1177.
  * [7] C.A.Robinson, Spectra of derived module homomorphisms, Math Proc Camb Phil Soc 101 (1987) 249–257.
  * [8] C.A.Weibel, An introduction to homological algebra, Cambridge studies in advanced mathematics 38 (1994).
  * [9] W.S.Wilson, A new relation on the Stiefel-Whitney classes of Spin manifolds, Ill Jour Math 17 (1973) 115–127.