# Octonion algebras over schemes and the equivalence of isotopes and isometric
quadratic forms111This work was done while the author was a master student at
the Department of Mathematics, Uppsala University. Supervisor was S. Alsaody.

V. Hildebrandsson222Department of Mathematics (MAI), Linköpings University,
581 83 Linköping, Sweden. Email address<EMAIL_ADDRESS>
###### Abstract

Octonion algebras are certain algebras with a multiplicative quadratic form.
In [1], Alsaody and Gille show that, for octonion algebras over unital
commutative rings, there is an equivalence between isotopes and isometric
quadratic forms. The contravariant equivalence from unital commutative rings
to affine schemes leads us to a question: can the equivalence of isometry and
isotopy be generalized to octonion algebras over a (not necessarily affine)
scheme? We present the basic definitions and properties of octonion algebras,
both over rings and over schemes. Then we show that an isotope of an octonion
algebra $\mathcal{C}$ over a scheme is isomorphic to a twist by an
$\mathbf{Aut}(\mathcal{C})$–torsor. We conclude the paper by giving an
affirmative answer to our question.

###### Contents

  1. 1 Introduction
  2. 2 Background
    1. 2.1 Sites and sheaves
    2. 2.2 Octonion algebras over rings
  3. 3 Octonion algebras over schemes
  4. 4 Twists, isotopes, and isometries
    1. 4.1 The twisted octonion algebra
    2. 4.2 The equivalence of isotopes and isometries

##  1 Introduction

The octonions $\mathbb{O}$ are a normed division algebra over $\mathbb{R}$ of
dimension 8, discovered in 1843 by John Graves. Based on these, the definition
of a octonion algebras over any field was given by Leonard Dickson in 1927,
and this was generalized once more to octonion algebras over rings by
Loos–Petersson–Racine in 2008 [9]. An important part of these definitions is
the assumption that the algebra has an associated quadratic form. In the case
over a field, an octonion algebra is completely determined by its quadratic
form [2, Claim 2.3]. Over rings however, this is not the case, proved by Gille
in [7]. The main result of Alsaody and Gille gives a construction of the
non–isomorphic octonion algebras with isometric quadratic forms. By the
contravariant equivalence of the category of rings and the category of affine
schemes, we expect this construction to generalize to octonion algebras over
affine schemes. However, will the construction generalize to octonion algebras
over general schemes?

Composition algebras over locally ringed spaces were first defined by
Petersson in [10]. Since a locally ringed space $(X,\mathcal{O}_{X})$ comes
with a sheaf of rings $\mathcal{O}_{X}$, the idea is to define the composition
algebra as a pair $(\mathscr{C},N)$ where $\mathscr{C}$ is an
$\mathcal{O}_{X}$–algebra that is locally a composition algebra in the ring
sense. Composition algebras only exist in ranks 1, 2, 4, 8, and a composition
algebra of rank 8 is called an octonion algebra.

The structure of the article is as follows. In Section 2, we recall background
of sheaf theory over general sites, the fppf site, and basic notions of
octonion algebras over rings. We also present the main result of Alsaody and
Gille. In Section 3, we give the corresponding definitions for octonion
algebras over schemes. In Section 4, we prove several analogous results of
those proved in [1, §4 and §6]. The main results are (1) twisting an octonion
algebra $\mathcal{C}$ over a scheme by an $\mathbf{Aut}(\mathcal{C})$–torsor
corresponds naturally to an isotope of $\mathcal{C}$, and (2) two octonion
algebras over schemes are isotopic if and only if their quadratic forms are
isometric.

### Acknowledgments

I want to thank Susanne Pumplün for enlightening me of the work of Petersson.
I also want to thank Philippe Gille for our discussion which helped improve
the thesis. Lastly, I am grateful to Seidon Alsaody who introduced me to this
problem and who helped me immensely, both throughout the master thesis project
and when rewriting it into this article.

##  2 Background

In this section, we will present some background on sheaf theory on general
sites and the site $(\mathbf{Sch}/X)_{\text{fppf}}$. We will assume knowledge
in standard sheaf theory (i.e. over the Zariski site) and in basic schemes
theory. We will also present theory of octonion algebras over rings and the
main result of Alsaody and Gille. For more on sheaf theory on general sites,
see [11, §7]. For more on the fppf site, see [11, §34.7]. For more on octonion
algebras over rings and proofs of results in Section 2.2, see [1, §2 and §6].

Throughout, if not stated otherwise, $R$ is a commutative and unital ring and
$X$ is a scheme. By an _$R$ –ring_ we mean a unital, commutative and
associative $R$–algebra. We denote the category of schemes by $\mathbf{Sch}$.

A _$X$ –scheme_ is a scheme $Y$ together with a morphism of schemes $Y\to X$.
A $X$–morphism $Y\to Z$ is a morphism of schemes such that the diagram

${Y}$${Z}$${X}$

commutes. We denote the category of $X$–schemes by $\mathbf{Sch}/X$.

By the Yoneda lemma, we can identify a scheme $Y\in\mathbf{Sch}$ (or
$X$–scheme $Y\in\mathbf{Sch}/X$) with its representable functor

$h_{Y}:\mathbf{Sch}\to\mathbf{Set}\quad(\text{or
}h_{Y}:\mathbf{Sch}/X\to\mathbf{Set}).$

We use the notation $Y(T):=h_{Y}(T)$.

A _group scheme_ (or $X$–group scheme) is a scheme (or $X$–scheme) $G$ and a
factorization of its representable functor $h_{G}$ through the forgetful
functor of groups to sets.

###  2.1 Sites and sheaves

In order to avoid set theoretical problems, we work over a fixed Grothendieck
universe.

A _site_ is a category $\mathscr{C}$ and a class $\mathbf{Cov}(\mathscr{C})$
of families of morphisms with fixed target $\\{U_{i}\to U\\}_{i\in I}$, called
coverings of $\mathscr{C}$, satisfying certain conditions, see [11, Definition
7.6.2]. We will denote the site by its category $\mathscr{C}$.

######  Example 2.1 (Fppf site).

An _fppf covering_ of $X$ is a family of morphisms with fixed target in
$\mathbf{Sch}$, $\\{p_{i}:X_{i}\to X\\}_{i\in I}$, such that $X=\bigcup_{i\in
I}p_{i}(X_{i})$ and each $p_{i}$ is flat and of locally finite presentation.
These fppf coverings makes $\mathbf{Sch}$ into a site [11, Lemma 34.7.3],
denoted by $\mathbf{Sch}_{\text{fppf}}$. We make $\mathbf{Sch}/X$ into a site
by defining

$(\mathbf{Sch}/X)_{\text{fppf}}:=(\mathbf{Sch}_{\text{fppf}})/X.$

In other words, a covering of $Y\in\mathbf{Sch}/X$ is an fppf covering
$\\{Y_{i}\to Y\\}_{i\in I}$, where $Y_{i}\in\mathbf{Sch}/X$, such that the
diagram

${Y_{i}}$${Y}$${X}$

commutes for all $i\in I$.

Let $\mathscr{C}$ be a category. A _presheaf_ $\mathcal{F}$ on $\mathscr{C}$
is a contravariant functor $\mathcal{F}:\mathscr{C}\to\mathbf{Set}$. If
$\mathscr{C}$ is a site, a presheaf $\mathcal{F}$ is a _sheaf_ if for every
covering $\\{U_{i}\to U\\}_{i\in I}\in\mathbf{Cov}(\mathscr{C})$ the following
holds:

  1. 1.

if two sections $s,t\in\mathcal{F}(U)$ are such that $\forall i\in I$
$s|_{U_{i}}=t|_{U_{i}}$ then $s=t$,

  2. 2.

if $\forall i\in I$ $s_{i}\in\mathcal{F}(U_{i})$ are such that $\forall j\in
I$ $s_{i}|_{U_{i}\times_{U}U_{j}}=s_{j}|_{U_{i}\times_{U}U_{j}}$, then there
exists $s\in\mathcal{F}(U)$ such that $\forall i\in I$ $s|_{U_{i}}=s_{i}$.

Let $\mathcal{F}$ and $\mathcal{G}$ be two presheaves. A _morphism of
presheaves_ is a natural transformation $\eta:\mathcal{F}\to\mathcal{G}$. If
$\mathcal{F}$ and $\mathcal{G}$ are sheaves, then $\eta$ is a _morphism of
sheaves_. The categories of presheaves and sheaves are denoted by
$\mathbf{PSh}(\mathscr{C})$ and $\mathbf{Sh}(\mathscr{C})$ respectively.

Let $\mathcal{F}$ be a presheaf on the site $\mathscr{C}$. A sheafification
functor $L:\mathbf{PSh}(\mathscr{C})\to\mathbf{Sh}(\mathscr{C})$ is a left
adjoint to the inclusion
$\mathbf{Sh}(\mathscr{C})\hookrightarrow\mathbf{PSh}(\mathscr{C})$. The
_sheafification of $\mathcal{F}$_ is then
$\mathcal{F}^{\prime}:=L(\mathcal{F})$ together with the canonical map
$\mathcal{F}\to\mathcal{F}^{\prime}$. For its construction, see [11, §7.10].

Let $\mathscr{C}$ be a site and $\mathcal{G}$ be a sheaf of groups on
$\mathscr{C}$. A _$\mathcal{G}$ –torsor_ is a sheaf of sets $\mathcal{F}$ on
$\mathscr{C}$ endowed with a $\mathcal{G}$–action
$\mathcal{G}\times\mathcal{F}\to\mathcal{F}$ such that

  1. 1.

whenever $\mathcal{F}(U)$ is non-empty the action
$\mathcal{G}(U)\times\mathcal{F}(U)\to\mathcal{F}(U)$ is transitive,

  2. 2.

for every $U\in\mathscr{C}$ there exists a covering $\\{U_{i}\to U\\}_{i\in
I}\in\mathbf{Cov}(\mathscr{C})$ such that $\mathcal{F}(U_{i})$ is non-empty
for all $i\in I$.

A morphism of $\mathcal{G}$–torsors is a morphism of sheaves compatible with
the $\mathcal{G}$–action.

###  2.2 Octonion algebras over rings

Let $A=(A,\ast)$ be an algebra. It is _alternative_ if for each $a,b\in A$

$a(ab)=(aa)b\text{ and }(ba)a=b(aa).$

For each $a\in A$ we have the linear maps $L_{a}:x\mapsto a\ast x$ and
$R_{a}:x\mapsto x\ast a$. If $A$ is alternative then $L_{a}R_{a}=R_{a}L_{a}$
for all $a\in A$, and we denote this map $B_{a}$.

Two $R$–algebras $(A,\ast_{A})$ and $(B,\ast_{B})$ are _isotopic_ if there
exist invertible linear maps $f_{i}:A\to B$, $i=1,2,3$, such that

$f_{1}(x\ast_{A}y)=f_{2}(x)\ast_{B}f_{3}(y)$

for all $x,y\in A$.

######  Remark 2.2.

If $A$ and $B$ are isotopic, then $A$ and $B$ are isomorphic as $R$–modules,
but not necessarily as $R$–algebras. However, $f_{1}:A\to B$ is an isomorphism
of algebras $A$ and $(B,\ast^{\prime}_{B})$, where

$x\ast^{\prime}_{B}y=f_{2}f_{1}^{-1}(x)\ast_{B}f_{3}f_{1}^{-1}(y).$

An algebra $(B,\ast_{B}^{\prime})$ is a _principal isotope_ of $(B,\ast_{B})$
if there exist invertible linear maps $g,h:B\to B$ such that
$x\ast^{\prime}y=g(x)\ast h(y)$ for all $x,y\in B$. We denote
$(B,\ast_{B}^{\prime})=:B_{g,h}$. The algebras $B$ and $B_{g,h}$ are obviously
isotopic. It follows from remark 2.2 that algebras $A$ and $B$ are isotopic if
and only if $A$ is isomorphic to a principal isotope of $B$.

Let $(A,\ast)$ be an $R$–algebra and $g,h:A\to A$ invertible maps. $A_{g,h}$
is unital if there exists $e\in A$ such that for all $x\in A$

$g(x)\ast h(e)=g(e)\ast h(x)=x.$

Equivalently,

$g^{-1}=R_{h(e)}\text{ and }h^{-1}=L_{g(e)}.$

If $A$ is a unital, alternative algebra, this implies that $g(e)$ and $h(e)$
are invertible elements (the inverse of an element is well defined in an
alternative algebra). Let $a=g(e)^{-1}$ and $b=h(e)^{-1}$. Then (2.2) is
equivalent to

$g=R_{a}\text{ and }h=L_{b}.$

Also, for any $a,b\in A^{\ast}$, $A_{R_{a},L_{b}}$ is unital with unity
$(ab)^{-1}$. Here $A^{\ast}$ denotes the invertible elements of $A$.

######  Proposition 2.3 (Alsaody and Gille, Proposition 2.1).

Let $A$ be a unital alternative algebra over $R$ and let $A^{\prime}$ be
isotopic to $A$. Then $A^{\prime}$ is unital if and only if $A^{\prime}\simeq
A_{R_{a},L_{b}}$ for some $a,b\in A^{\ast}$.

######  Remark 2.4.

  1. 1.

We denote $A_{R_{a},L_{b}}=:A^{a,b}$.

  2. 2.

If $\phi:R\to S$ is a ring morphism, we can pullback an $R$–algebra $A$ to an
$S$–algebra $A_{S}:=A\otimes_{R}S$. For $a,b\in A^{\ast}$, we have
$(A^{a,b})_{S}=(A_{S})^{a_{S},b_{S}}$. We denote this algebra by
$A^{a,b}_{S}$.

A quadratic form on an $R$–module $M$ is a map $q:M\to R$ such that

$q(rm)=r^{2}q(m),\quad r\in R,m\in M,$

and

$b_{q}(m,n):=q(m+n)-q(m)-q(n),\quad m,n\in M$

is bilinear. It is multiplicative if

$q(mn)=q(m)q(n),\quad m,n\in M,$

and regular if the morphism

$\displaystyle\alpha:M$ $\displaystyle\to\text{Hom}(M,R)$ $\displaystyle m$
$\displaystyle\mapsto b_{q}(m,\\_)$

is an isomorphism. If $M$ is finitely generated and projective, the rank of
$M$ at $p\in\mathbf{Spec}(R)$ is the rank of the free $R_{p}$–module $M_{p}$.
It is a locally constant function on $\mathbf{Spec}(R)$.

Now, an _octonion algebra over $R$_ is an $R$–algebra whose underlying module
is projective of constant rank 8, and which is endowed with a regular
multiplicative quadratic form. It is unital and alternative.

######  Example 2.5.

Over any ring $R$, we have the _Zorn algebra_

$\mathbf{Zorn}(R)=\begin{pmatrix}R&R^{3}\\\ R^{3}&R\end{pmatrix}$

with multiplication defined by

$\begin{pmatrix}a&u\\\ u^{\prime}&a^{\prime}\end{pmatrix}\begin{pmatrix}b&v\\\
v^{\prime}&b^{\prime}\end{pmatrix}=\begin{pmatrix}ab+u^{t}v^{\prime}&av+b^{\prime}u-u^{\prime}\times
v^{\prime}\\\ a^{\prime}v^{\prime}+bu^{\prime}+u\times
v&(u^{\prime})^{t}v+a^{\prime}b^{\prime}\end{pmatrix}$

where $u\times v$ is the vector cross product and $u^{t}v$ is the scalar
product. With the quadratic form given by the determinant

$\det\begin{pmatrix}a&u\\\
u^{\prime}&a^{\prime}\end{pmatrix}=aa^{\prime}-u^{t}u^{\prime},$

$\mathbf{Zorn}(R)$ becomes an $R$–octonion algebra.

######  Remark 2.6.

  1. 1.

If $(C,q)$ is an octonion algebra, we have the equality $C^{\ast}=\\{x\in
C|q(x)\in R^{\ast}\\}$.

  2. 2.

The trace of a quadratic form $q$ is the map $tr:M\to R,m\mapsto b_{q}(1,m)$.
Then we define the involution map by $\kappa:x\mapsto\overline{x}:=tr(x)\cdot
1_{C}-x$.

  3. 3.

For any octonion algebra $(C,q)$ over $R$, we have the _Hamilton–Cayley
equation_ $q(c)1=c\overline{c}$, for any $c\in C$ [10].

For any octonion algebra $C$ over $R$ we have the octonion sphere
$\mathbf{S}_{C}$, which is the $R$–scheme

$\mathbf{S}_{C}(S)=\\{c\in C_{S}|q_{C_{S}}(c)=1\\}.$

We can now state the result of Alsaody and Gille.

######  Theorem 2.7 (Alsaody and Gille, Corollary 6.7).

Let $C$ and $C^{\prime}$ be octonion algebras over $R$. The quadratic forms
$q_{C}$ and $q_{C^{\prime}}$ are isometric if and only if there exist
$a,b\in\mathbf{S}_{C}(R)$ such that $C^{\prime}\simeq C^{a,b}$.

##  3 Octonion algebras over schemes

In this section, we will present the definition of an octonion algebra over a
scheme and some of its constructions. These definitions will be
generalizations of the ones defined in subsection 2.2.

######  Definition 3.1 (Octonion algebra over a scheme).

An _octonion algebra_ over $X$ is a tuple $(\mathcal{C},\mathcal{Q})$
consisting of

  1. 1.

a sheaf $\mathcal{C}$ of $\mathcal{O}_{X}$-algebras such that for all open
affine $U\subset X$, the underlying module of the $\mathcal{O}_{X}(U)$-algebra
$\mathcal{C}(U)$ is projective with constant rank 8,

  2. 2.

a morphism of sheaves $\mathcal{Q}:\mathcal{C}\to\mathcal{O}_{X}$ such that
for all open affine $U\subset X$, $\mathcal{Q}(U)$ is a multiplicative,
regular, quadratic form on the $\mathcal{O}_{X}(U)$–algebra $\mathcal{C}(U)$.
We call $\mathcal{Q}$ the _quadratic form on $\mathcal{C}$_.

######  Remark 3.2.

In the case where $X=\mathbf{Spec}(R)$ is an affine scheme,
$\mathcal{O}_{X}(X)=R$, so for an octonion algebra $\mathcal{C}$ over $X$,
$\mathcal{C}(X)$ is an $R$-octonion algebra (as defined in subsection 2.2).

Note that defining the stalks and induced stalk maps
$(\mathcal{C}_{x},\mathcal{Q}_{x})$ to be octonion algebras over
$\mathcal{O}_{X,x}$, as defined in [10, §1.6 and §1.7], is another natural way
to define the octonion algebra over a scheme. In the following proposition and
corollary, we see that the definitions are equivalent.

######  Proposition 3.3.

Let $\mathcal{Q}:\mathcal{C}\to\mathcal{O}_{X}$ be a natural transformation.
Then, for some open affine set $U\subset X$, $\mathcal{Q}(U)$ is a regular,
multiplicative, and quadratic form if and only if $\mathcal{Q}_{x}$ is a
regular, multiplicative, and quadratic form for all $x\in U$.

###### Proof.

We only show the multiplicative part, regularity and being quadratic is
similar. The induced stalk map $\mathcal{Q}_{x}$ at $x\in U$ can be defined in
two steps:

$\displaystyle 1.$
$\displaystyle\quad\tilde{\mathcal{Q}}_{x}:(U,s)\mapsto(U,\mathcal{Q}(U)(s))$
$\displaystyle 2.$
$\displaystyle\quad\mathcal{Q}_{x}:\overline{(U,s)}\mapsto\overline{\tilde{\mathcal{Q}}_{x}(U,s)}.$

Assume $\mathcal{Q}(U)$ is a regular, multiplicative, and quadratic form.
Then, by the definition of $\mathcal{Q}_{x}$ for $x\in U$, it follows that
$\mathcal{Q}_{x}$ is aswell.

Now, assume for all $x\in U$ that $\mathcal{Q}_{x}$ is multiplicative. Then
$\tilde{\mathcal{Q}}_{x}$ must be multiplicative. For $a,b\in\mathcal{C}(U)$,
we have

$\displaystyle(U,\mathcal{Q}(U)(ab))$
$\displaystyle=\tilde{\mathcal{Q}}_{x}((U,ab))$
$\displaystyle=\tilde{\mathcal{Q}}_{x}((U,a)(U,b))$
$\displaystyle=\tilde{\mathcal{Q}}_{x}((U,a))\tilde{\mathcal{Q}}_{x}((U,b))$
$\displaystyle=(U,\mathcal{Q}(U)(a))(U,\mathcal{Q}(U)(b))$
$\displaystyle=(U,\mathcal{Q}(U)(a)\mathcal{Q}(U)(b))$

so $\mathcal{Q}(U)$ must also be multiplicative. The proof of regularity and
being quadratic is similar. ∎

######  Corollary 3.4.

For some open affine set $U\subset X$, $(\mathcal{C}(U),\mathcal{Q}(U))$ is an
octonion algebra over $\mathcal{O}_{X}(U)$ if and only if
$(\mathcal{C}_{x},\mathcal{Q}_{x})$ is an octonion algebra over
$\mathcal{O}_{X,x}$ for all $x\in U$.

###### Proof.

The rank of $\mathcal{C}(U)$ at $x$ is by definition the rank of
$\mathcal{C}_{x}$, and $\mathcal{C}(U)$ is projective if and only if
$\mathcal{C}_{x}$ is projective for all $x\in U$ [6, Exercise 4.11b]. Then the
statement follows from proposition 3.3. ∎

Let $(\mathcal{C},\mathcal{Q})$ and
$(\mathcal{C}^{\prime},\mathcal{Q}^{\prime})$ be two octonion algebras over
$X$. A _morphism_
$\varphi:(\mathcal{C},\mathcal{Q})\to(\mathcal{C}^{\prime},\mathcal{Q}^{\prime})$
is a natural transformation $\varphi:\mathcal{C}\to\mathcal{C}^{\prime}$ such
that $\varphi(U)$ is an algebra morphism for all open sets $U\subset X$.

######  Remark 3.5.

If we have an isomorphism
$\varphi:(\mathcal{C}^{\prime},\mathcal{Q}^{\prime})\to(\mathcal{C},\mathcal{Q})$
of octonion algebras over $X$, we also have an isomorphism of octonion
algebras
$(\mathcal{C}(U),\mathcal{Q}(U))\simeq(\mathcal{C}^{\prime}(U),\mathcal{Q}^{\prime}(U))$
over the ring $\mathcal{O}_{X}(U)$. It follows that $\mathcal{Q}(U)$ and
$\mathcal{Q}^{\prime}(U)$ are isometric, i.e.

$\mathcal{Q}^{\prime}(U)=\mathcal{Q}(U)\circ\varphi(U)=(\mathcal{Q}\circ\varphi)(U).$

Since this holds for all open affine $U\subset X$, and $X$ has an open affine
cover, it follows that $\mathcal{Q}^{\prime}=\mathcal{Q}\circ\varphi$, i.e.
$\mathcal{Q}$ and $\mathcal{Q}^{\prime}$ are isometric. We denote that two
quadratic forms are isometric by $\mathcal{Q}\sim\mathcal{Q}^{\prime}$.

Let $(\mathcal{C},\mathcal{Q})$ be an octonion algebra over $X$. For an
octonion algebra over a ring, the form induces a bilinear form, a trace, and a
natural involution. The same goes for the quadratic form $\mathcal{Q}$. The
bilinear form is the morphism of sheaves
$\mathcal{B_{Q}}:\mathcal{C}\times\mathcal{C}\to\mathcal{O}_{X}$ defined for
each open set $U\subset X$ by

$\mathcal{B_{Q}}(U):(c,c^{\prime})\mapsto\mathcal{Q}(U)(c+c^{\prime})-\mathcal{Q}(U)(c)-\mathcal{Q}(U)(c^{\prime}).$

The trace is then given by the morphism of sheaves
$\text{tr}_{\mathcal{C}}:\mathcal{C}\to\mathcal{O}_{X}$ defined for each open
set $U\subset X$ by

$\text{tr}_{\mathcal{C}}(U):c\mapsto\mathcal{B_{Q}}(U)(1,c)$

and the natural involution is the morphism of sheaves
$\kappa_{\mathcal{C}}:\mathcal{C}\to\mathcal{C}$ defined for each open set
$U\subset X$ by

$\kappa_{\mathcal{C}}(U):c\mapsto\text{tr}_{\mathcal{C}}(U)(c)\cdot
1_{\mathcal{C}(U)}-c=:\overline{c}$

Now, let $a\in\mathcal{C}(X)$ be a global section. Let
$L_{a}:\mathcal{C}\to\mathcal{C}$ and $R_{a}:\mathcal{C}\to\mathcal{C}$ be the
morphisms of sheaves defined for any open $U\subset X$ by

$\displaystyle L_{a}(U):x\mapsto\text{res}_{X,U}(a)\cdot x,\quad
R_{a}(U):x\mapsto x\cdot\text{res}_{X,U}(a).$

Also, let $B_{a}:=R_{a}L_{a}$.

The _octonion unit sphere_ is the $X$–scheme defined by

$\displaystyle\mathbf{S}_{\mathcal{Q_{C}}}:\mathbf{Sch}/X$
$\displaystyle\to\mathbf{Set}$ $\displaystyle Y$
$\displaystyle\mapsto\\{c\in\mathcal{C}_{Y}(Y)|\mathcal{Q}_{\mathcal{C}_{Y}}(Y)(c)=1\\}.$

######  Definition 3.6 (Isotope).

Let $\mathcal{C}$ be an octonion algebra over $X$. For each pair
$a,b\in\mathbf{S}_{\mathcal{Q_{C}}}(X)$, we have an _isotope of_
$\mathcal{C}$, denoted $\mathcal{C}^{a,b}$, which is the sheaf of
$\mathcal{O}_{X}$-algebras such that, for any open $U\subset X$,
$(\mathcal{C}^{a,b}(U),\ast)$ is the $\mathcal{O}_{X}(U)$-algebra with
multiplication

$x\ast y:=R_{a}(U)(x)\cdot L_{b}(U)(y).$

######  Remark 3.7.

Using the quadratic form $\mathcal{Q}$ of $\mathcal{C}$, the isotope
$\mathcal{C}^{a,b}$ becomes an octonion algebra over $X$. Also, it follows
that $(\mathcal{C}^{a,b}(U),\mathcal{Q}(U))$ is an isotope of
$(\mathcal{C}(U),\mathcal{Q}(U))$ as $\mathcal{O}_{X}(U)$–octonion algebras,
for all open affine $U\subset X$. By this and remark 3.2, we see that the
definitions of an octonion algebra over a scheme and of an isotope are natural
generalizations of those over rings, as we wanted.

Now, let $Y$ be an $X$–scheme. We make an octonion algebra $\mathcal{C}$ over
$X$ into an octonion algebra $\mathcal{C}_{Y}$ over $Y$. Consider the presheaf

$V\mapsto(f^{-1}\mathcal{C})(V)\otimes_{(f^{-1}\mathcal{O}_{X})(V)}\mathcal{O}_{Y}(V),$

and let $\mathcal{C}_{Y}$ be its sheafification (with respect to the Zariski
site). Then $\mathcal{C}_{Y}$, together with
$\mathcal{Q}_{Y}:=f^{-1}\mathcal{Q}\otimes id$, where $f^{-1}\mathcal{Q}$ is
the inverse sheaf functor acting on the sheaf morphism $\mathcal{Q}$, is an
octonion algebra over $Y$ [10, Proposition 1.7]. (The motivation is the
construction in remark 2.4. If $X=\mathbf{Spec}(R)$ and $Y=\mathbf{Spec}(S)$,
we get exactly $\mathcal{C}_{Y}(Y)=C(X)_{S}$.)

######  Remark 3.8.

  1. 1.

Let $U\subset X$ be an open set. For a section $a\in\mathcal{C}(U)$ of an
octonion algebra $\mathcal{C}$ over $X$, we have, from the definition of
$\mathcal{C}_{Y}$, the associated section $a_{Y}:=[a]\otimes
1\in\mathcal{C}_{Y}(f^{-1}(U))$, where $[a]$ is the equivalence class of $a$
in $(f^{-1}\mathcal{C})(f^{-1}(U))$. In particular, if $a\in\mathcal{C}(X)$ is
a global section, $a_{Y}\in\mathcal{C}_{Y}(Y)$ is a global section.

  2. 2.

Let $V\subset Y$ be an open set. In both $(\mathcal{C}_{Y})^{a_{Y},b_{Y}}(V)$
and $(\mathcal{C}^{a,b})_{Y}(V)$, the elements are of the form $c_{Y}$, for
some $c\in\mathcal{C}(U)$, $fV\subset U\subset X$, where $U$ is open. Elements
$c_{Y},d_{Y}\in(\mathcal{C}_{Y})^{a_{Y},b_{Y}}$ are multiplied as

$c_{Y}\ast_{1}d_{Y}=\big{(}c_{Y}\cdot\text{res}_{Y,V}(a_{Y})\big{)}\big{(}\text{res}_{Y,V}(b_{Y})\cdot
d_{Y}\big{)},$

while $c_{Y},d_{Y}\in(\mathcal{C}^{a,b})_{Y}$ are multiplied as

$c_{Y}\ast_{2}d_{Y}=\big{[}((c\cdot\text{res}_{X,U}(a))(\text{res}_{X,U}(b)\cdot
d))\big{]}\otimes 1.$

However,

$\displaystyle c_{Y}\ast_{2}d_{Y}$
$\displaystyle=\big{[}((c\cdot\text{res}_{X,U}(a))(\text{res}_{X,U}(b)\cdot
d))\big{]}\otimes 1$
$\displaystyle=\big{[}(c\cdot\text{res}_{X,U}(a))\big{]}\otimes
1\cdot\big{[}(\text{res}_{X,U}(b)\cdot d)\big{]}\otimes 1$
$\displaystyle=\big{(}[c]\otimes 1\cdot[\text{res}_{X,U}(a)]\otimes
1\big{)}\cdot\big{(}[\text{res}_{X,U}(b)]\otimes 1\cdot[d]\otimes 1\big{)}$
$\displaystyle=\big{(}c_{Y}\cdot\text{res}_{Y,V}(a_{Y})\big{)}\big{(}\text{res}_{Y,V}(b_{Y})\cdot
d_{Y}\big{)}$ $\displaystyle=c_{Y}\ast_{1}d_{Y},$

by naturality of the restriction maps. So
$(\mathcal{C}_{Y})^{a_{Y},b_{Y}}=(\mathcal{C}^{a,b})_{Y}$, and we denote this
sheaf by $\mathcal{C}_{Y}^{a,b}$.

######  Example 3.9.

Recall that over $\mathbb{Z}$, we have the Zorn algebra
$\mathbf{Zorn}(\mathbb{Z})$. By the contravariant equivalence of
$\mathbf{Ring}$ and $\mathbf{Aff}$, $\mathbf{Zorn}(\mathbb{Z})$ defines an
octonion algebra over $\mathbf{Spec}(\mathbb{Z})$ as well. Let $Y$ be a
scheme. By the pullback construction, and the fact that any scheme is a
$\mathbb{Z}$–scheme, $\mathbf{Zorn}(\mathbb{Z})_{Y}$ exists and is an octonion
algebra over $Y$.

##  4 Twists, isotopes, and isometries

In this section, we will prove the generalizations of four results in [1]
(theorems 4.1, 4.6, and 6.6, and corollary 6.7, corresponding to theorems 4.3,
4.7, and 4.12, and corollary 4.13 in this article).

###  4.1 The twisted octonion algebra

Let us start with the affine case. We want to define the $R$–group schemes
$\mathbf{Aut}(\mathcal{C})$, $\mathbf{SO}(q_{C})$, and $\mathbf{RT}(C)$. Let
$(C,q_{C})$ be an octonion $R$–algebra. Denote by $\mathbf{Aut}(C)$ the
automorphism $R$–group scheme of $C$, defined, for any $R$–ring $S$ by

$\mathbf{Aut}(C)(S):=\mathbf{Aut}(C_{S}),$

which is a closed subscheme of the $R$–group scheme $\mathbf{GL}(C)$. By
$\mathbf{SO}(q_{C})$ we denote the special orthogonal $R$–group scheme
associated to $q_{C}$ [3, Section 4.3]. The embedding
$\mathbf{Aut}(C)\to\mathbf{GL}(C)$ induces a closed immersion
$\mathbf{Aut}(C)\to\mathbf{SO}(q_{C})$. Define $\mathbf{RT}(C)$ as the closed
$R$–subscheme of $\mathbf{SO}(q_{C})^{3}$ such that for any $R$–ring $S$ we
have

$\mathbf{RT}(C)(S)=\left\\{(t_{1},t_{2},t_{3})\in\mathbf{SO}(q_{C})^{3}|t_{1}(xy)=\overline{t_{2}(\overline{x})}\cdot\overline{t_{3}(\overline{y})}\text{
for any }x,y\in C_{S}\right\\}.$

Now we can continue with octonion algebras over general schemes. Let $X$ be a
scheme. From here onwards, unless otherwise stated, we denote an $X$–scheme
$Y\overset{f}{\to}X$ by its scheme $Y$. Let $(\mathcal{C},\mathcal{Q_{C}})$ be
an octonion algebra over $X$. Let $\mathbf{Aut}(\mathcal{C})$ be the affine
group scheme over $X$ such that

$\mathbf{Aut}(\mathcal{C})(Y):=\mathbf{Aut}(\mathcal{C}_{Y}).$

The $X$–group scheme $\mathbf{O}(\mathcal{Q}_{\mathcal{C}})$ is defined by

$\displaystyle\mathbf{O}(\mathcal{Q}_{\mathcal{C}})(Y):=\\{\varphi:\mathcal{C}_{Y}\to\mathcal{C}_{Y}|$
$\displaystyle\varphi_{Z}:(\mathcal{C}_{Y})_{Z}\to(\mathcal{C}_{Y})_{Z}\text{
is linear, invertible,}$ $\displaystyle\text{and
}\mathcal{Q}_{(\mathcal{C}_{Y})_{Z}}\circ\varphi=\mathcal{Q}_{(\mathcal{C}_{Y})_{Z}}\text{
for any $Y$–scheme }Z\\}$

[3, Section 4.1] and, similar to the affine case,
$\mathbf{SO}(\mathcal{Q}_{\mathcal{C}})$ is the special orthogonal $X$–group
scheme [3, Section 4.3]. Recall that for $c\in\mathcal{C}(U)$, we have the
natural involution
$\kappa_{\mathcal{C}}(U)(c)=\text{tr}_{\mathcal{C}}(U)(c)-c$. We then define
$\mathbf{RT}(\mathcal{C})$ as the closed $X$–subscheme of
$\mathbf{SO}(\mathcal{Q}_{\mathcal{C}})^{3}$ defined by

$\displaystyle\mathbf{RT}(\mathcal{C})(Y):=\\{(t_{1},t_{2},t_{3})\in\mathbf{SO}$
$\displaystyle(\mathcal{Q}_{\mathcal{C}})(Y)^{3}\big{|}\text{ for any
$Y$–scheme }Z$ $\displaystyle
t_{1Z}(Z)(xy)=(\kappa_{(\mathcal{C}_{Y})_{Z}}t_{2Z}\kappa_{(\mathcal{C}_{Y})_{Z}})(Z)(x)\cdot(\kappa_{(\mathcal{C}_{Y})_{Z}}t_{3Z}\kappa_{(\mathcal{C}_{Y})_{Z}})(Z)(y)\\}.$

where $t_{iZ}$, $i=1,2,3$, are induced by the $Y$–scheme structure morphism
$Z\to Y$.

######  Remark 4.1.

These schemes can equivalently be defined in with a global–local point of
view:

$\mathbf{O}(\mathcal{Q}_{\mathcal{C}})(Y)=\\{\varphi:\mathcal{C}_{Y}\to\mathcal{C}_{Y}|\text{
for all open affine }V\subset
Y;\varphi(V)\in\mathbf{O}(\mathcal{Q}_{\mathcal{C}_{Y}}(V))\\},$
$\mathbf{SO}(\mathcal{Q}_{\mathcal{C}})(Y)=\\{\varphi\in\mathbf{O}(\mathcal{Q}_{\mathcal{C}})(Y)|\text{
for all open affine }V\subset
Y;\varphi(V)\in\mathbf{SO}(\mathcal{Q}_{\mathcal{C}_{Y}}(V))\\},$
$\displaystyle\mathbf{RT}(\mathcal{C})(Y)=\\{(t_{1},t_{2},t_{3})\in\mathbf{SO}(\mathcal{Q}_{\mathcal{C}})(Y)^{3}|$
$\displaystyle\text{ for all open affine }V\subset Y$
$\displaystyle(t_{1}(V),t_{2}(V),t_{3}(V))\in\mathbf{RT}(\mathcal{C}_{Y}(V))(\mathcal{O}_{Y}(V))\\},$

From [1], we know many properties of these group schemes over rings ($\simeq$
affine schemes). We will see that this perspective will prove useful.

######  Example 4.2.

We see that, for any affine open $U\subset X$, $a\in\mathcal{C}(X)$,
$x,y\in\mathcal{C}(U)$, the triple $(B_{a},R_{\overline{a}},L_{\overline{a}})$
satisfies

$\displaystyle B_{a}(U)(xy)$ $\displaystyle=R_{a}(U)L_{a}(U)(xy)$
$\displaystyle=\text{res}_{X,U}(a)\cdot((xy)\cdot\text{res}_{X,U}(a))$
$\displaystyle=(\overline{\text{res}_{X,U}(\overline{a})\cdot\overline{x}})\cdot(\overline{\overline{y}\cdot\text{res}_{X,U}(\overline{a})})$
$\displaystyle=R_{\overline{a}}(U)(x)L_{\overline{a}}(U)(y),$

where the third equality follows from the fourth Moufang identity. Hence
$(B_{a},R_{\overline{a}},L_{\overline{a}})\in\mathbf{RT}(\mathcal{C})(U)$.

We have a natural action of $\mathbf{RT}(\mathcal{C})$ on
$\mathbf{S}_{\mathcal{Q_{C}}}\times_{X}\mathbf{S}_{\mathcal{Q_{C}}}=:\mathbf{S}_{\mathcal{Q_{C}}}^{2}$
by

$(t_{1},t_{2},t_{3}).(u,v)=(t_{3}(Y)(u),t_{2}(Y)(v))$

for any $(t_{1},t_{2},t_{3})\in\mathbf{RT}(\mathcal{C})(Y)$ and any
$u,v\in\mathbf{S}_{\mathcal{Q_{C}}}(Y)$. We also have a map
$\Pi:\mathbf{RT}(\mathcal{C})\to\mathbf{S}_{\mathcal{Q_{C}}}^{2}$, defined for
any $\mathbf{t}:=(t_{1},t_{2},t_{3})\in\mathbf{RT}(\mathcal{C})(Y)$ by

$\Pi_{Y}:\mathbf{t}\mapsto(t_{3}(Y)(1),t_{2}(Y)(1)).$

######  Theorem 4.3.

  1. 1.

The stabilizer $\text{\emph{Stab}}_{\mathbf{RT}(\mathcal{C})}(1,1)$ is
$i(\mathbf{Aut}(\mathcal{C}))$, where
$i:\mathbf{Aut}(\mathcal{C})\hookrightarrow\mathbf{RT}(\mathcal{C})$ is the
natural transformation $i_{Y}:t\mapsto(t,t,t)$.

  2. 2.

The fppf quotient sheaf $\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})$
is representable by an $X$–scheme and the map

$\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})\to\mathbf{S}_{\mathcal{Q_{C}}}^{2},$

induced by $\Pi$, is an $X$–isomorphism.

###### Proof.

  1. 1.

First we have to show that $i$ is well-defined. Take
$t\in\mathbf{SO}(\mathcal{Q}_{\mathcal{C}})(Y)$ and open affine $V\subset Y$.
Then $t(V)$ is an automorphism if and only if

$(t(V),t(V),t(V))\in\mathbf{RT}(\mathcal{C}_{Y}(V))(\mathcal{O}_{Y}(V))$

[1, Lemma 3.5]. Hence $t\in\mathbf{Aut}(\mathcal{C})(Y)$ if and only if
$(t,t,t)\in\mathbf{RT}(\mathcal{C})(Y)$.

Now,
$(t_{1}(V),t_{2}(V),t_{3}(V))\in\mathbf{RT}(\mathcal{C}_{Y}(V))(\mathcal{O}_{Y}(V))$
satisfies

$t_{3}(V)(1)=t_{2}(V)(1)=1$

if and only if

$(t_{1}(V),t_{2}(V),t_{3}(V))\in
i_{V}\big{(}\mathbf{Aut}(\mathcal{C}_{Y}(V))\big{)}$

[1, Proposition 3.8]. Since it holds for all open affine $V\subset Y$, and $Y$
can be covered by open affines, we have that
Stab${}_{\mathbf{RT}(\mathcal{C})}(1,1)=i(\mathbf{Aut}(\mathcal{C}))$.

  2. 2.

We first want to show that $\mathbf{Aut}(\mathcal{C})$ is flat over $X$. Let
$h:\mathbf{Aut}(\mathcal{C})\to X$ be the canonical map, and
$X=\bigcup_{i}U_{i}$ an affine cover, $U_{i}=\mathbf{Spec}(R_{i})$ (note that,
by [11, Lemma 34.7.2], this cover is also an fppf covering). Since
$\mathbf{Aut}(\mathcal{C})$ is an affine $X$-scheme, $V_{i}:=h^{-1}(U_{i})$ is
an affine open set. $\mathbf{Aut}(\mathcal{C})|_{U_{i}}$ is an $R_{i}$–scheme,
and, by definition, $V_{i}$ is an open set of
$\mathbf{Aut}(\mathcal{C})|_{U_{i}}$. Then
$\mathcal{O}_{\mathbf{Aut}(\mathcal{C})|_{U_{i}}}(V_{i})$ is a flat
$R_{i}$–module [9, Corollary 4.12]. But
$\mathcal{O}_{\mathbf{Aut}(\mathcal{C})|_{U_{i}}}(V_{i})=\mathcal{O}_{\mathbf{Aut}(\mathcal{C})}(V_{i})$,
so $h$ is a flat morphism [11, Lemma 29.25.3]. Then
$\mathbf{Aut}(\mathcal{C})$ is flat over $X$. It follows that
$\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})$ is representable by an
$X$–scheme, and the induced map is a monomorphism [5, Expose XVI Theorem 2.2].

Let

$\displaystyle P:\mathbf{Sch}/X$ $\displaystyle\to\mathbf{Set}$ $\displaystyle
Z$
$\displaystyle\mapsto\mathbf{RT}(\mathcal{C})(Z)/\mathbf{Aut}(\mathcal{C})(Z)$

be a presheaf. Then
$\widetilde{P}:=\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})$ is its the
(fppf) sheafification of $P$. Let $\\{U_{i}\\}_{i\in I}$ be an affine cover of
$X$ and consider the presheaf

$\displaystyle Q_{i}:\mathbf{Sch}/U_{i}$ $\displaystyle\to\mathbf{Set}$
$\displaystyle Z$
$\displaystyle\mapsto\mathbf{RT}(\mathcal{C})|_{U_{i}}(Z)/\mathbf{Aut}(\mathcal{C})|_{U_{i}}(Z)$

Then
$\widetilde{Q_{i}}:=\mathbf{RT}(\mathcal{C})|_{U_{i}}/\mathbf{Aut}(\mathcal{C})|_{U_{i}}$
is its (fppf) sheafification. For any
$Z\in\mathbf{Sch}/U_{i}\subset\mathbf{Sch}/X$, the presheaves $P$ and $Q_{i}$
coincide:

$P(Z)=\mathbf{RT}(\mathcal{C})(Z)/\mathbf{Aut}(\mathcal{C})(Z)=\mathbf{RT}(\mathcal{C})|_{U_{i}}(Z)/\mathbf{Aut}(\mathcal{C})|_{U_{i}}(Z)=Q_{i}(Z).$

Then both $\widetilde{Q_{i}}$ and
$\Big{(}\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})\Big{)}\Big{|}_{U_{i}}$
satisfy the universal property of sheafification of $P|_{U_{i}}$. It follows
from uniqueness that

$\Big{(}\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})\Big{)}\Big{|}_{U_{i}}\simeq\widetilde{Q_{i}}=\mathbf{RT}(\mathcal{C})|_{U_{i}}/\mathbf{Aut}(\mathcal{C})|_{U_{i}}.$

Note also that, for any $Z\in\mathbf{Sch}/U_{i}\subset\mathbf{Sch}/X$, we also
have the equivalence

$\displaystyle(\mathbf{S}_{\mathcal{Q_{C}}}|_{U_{i}}\times_{U_{i}}\mathbf{S}_{\mathcal{Q_{C}}}|_{U_{i}})(Z)$
$\displaystyle=\mathbf{S}_{\mathcal{Q_{C}}}|_{U_{i}}(Z)\times_{U_{i}}\mathbf{S}_{\mathcal{Q_{C}}}|_{U_{i}}(Z)$
$\displaystyle=\mathbf{S}_{\mathcal{Q_{C}}}(Z)\times_{X}\mathbf{S}_{\mathcal{Q_{C}}}(Z)$
$\displaystyle=(\mathbf{S}_{\mathcal{Q_{C}}}\times_{X}\mathbf{S}_{\mathcal{Q_{C}}})(Z)$
$\displaystyle=(\mathbf{S}_{\mathcal{Q_{C}}}\times_{X}\mathbf{S}_{\mathcal{Q_{C}}})|_{U_{i}}(Z).$

Now, restricting the induced monomorphism we get

$\mathbf{RT}(\mathcal{C})|_{U_{i}}/\mathbf{Aut}(\mathcal{C})|_{U_{i}}\to\big{(}\mathbf{S}_{\mathcal{Q_{C}}}|_{U_{i}}\big{)}^{2},$

which is an $U_{i}$–isomorphism [1, Theorem 4.1(2)]. It follows that

$\mathbf{RT}(\mathcal{C})/\mathbf{Aut}(\mathcal{C})\overset{\sim}{\to}\mathbf{S}_{\mathcal{Q_{C}}}^{2}$

is an $X$–isomorphism.

∎

Let $X=\bigcup_{i}U_{i}$ be an affine cover and
$\mathbf{G}:=\mathbf{Aut}(\mathcal{C})$. It follows from theorem 4.3 and [4,
Corollary III.4.1.8] that $\Pi^{-1}|_{U_{i}}$ is a
$\mathbf{G}|_{U_{i}}$–torsor in the fppf topology. Then $\Pi^{-1}$ is a
$\mathbf{G}$–torsor over $\mathbf{S}_{\mathcal{Q_{C}}}^{2}$ in the fppf
topology.

We have the additive $X$–group scheme $\mathbf{W}(\mathcal{C})$ defined by

$\mathbf{W}(\mathcal{C})(Y)=\mathcal{C}_{Y}(Y).$

Given $a,b\in\mathbf{S}_{\mathcal{Q_{C}}}(Y)$, consider the
$\mathbf{G}$–torsor $\mathbf{E}^{a,b}:=\Pi^{-1}(a,b)$ over $X$. Let the
presheaf $\mathbf{P}^{a,b}(\mathcal{C})$ of $X$–schemes be definied by

$\mathbf{P}^{a,b}(\mathcal{C})(Y)=(\Pi^{-1}_{Y}(a_{Y},b_{Y})\times\mathcal{C}_{Y}(Y))/\sim$

where

$(\mathbf{t},x)\sim(\mathbf{t}^{\prime},x^{\prime})\iff\exists
g\in\mathbf{G}(Y):(\mathbf{t}g,x)=(\mathbf{t}^{\prime},g(Y)(x^{\prime})),$

and let $\mathbf{E}^{a,b}\wedge^{\mathbf{G}}\mathbf{W}(\mathcal{C})$ be its
sheafification.

######  Remark 4.4.

Let $\Pi_{Y}(\mathbf{t})=\Pi_{Y}(\mathbf{t}^{\prime})$. Then

$\displaystyle t_{3}(Y)(1)$ $\displaystyle=t_{3}^{\prime}(Y)(1)\iff
t_{3}^{-1}(Y)t_{3}^{\prime}(Y)(1)=1\iff(t_{3}^{-1}t_{3}^{\prime})(Y)(1)=1$
$\displaystyle t_{2}(Y)(1)$ $\displaystyle=t_{2}^{\prime}(Y)(1)\iff
t_{2}^{-1}(Y)t_{2}^{\prime}(Y)(1)=1\iff(t_{2}^{-1}t_{2}^{\prime})(Y)(1)=1$

which implies that
$\mathbf{t}^{-1}\mathbf{t}^{\prime}\in\mathbf{Stab}_{\mathbf{RT}(\mathcal{C})}(1,1)(Y)=i_{Y}(\mathbf{G}(Y))$.

The presheaf $\mathbf{P}^{a,b}(\mathcal{C})$ is in fact a presheaf of
algebras; the linear structure on $\mathbf{P}^{a,b}(\mathcal{C})(Y)$ is
induced by

$(\mathbf{t},x)+(\mathbf{t}^{\prime},x^{\prime})=(\mathbf{t},x+(t_{1}^{-1}t_{1}^{\prime})(Y)(x^{\prime}))$

and, for $s\in\mathcal{O}_{Y}(Y)$,

$s(\mathbf{t},x)=(\mathbf{t},sx),$

and the multiplication

$(\mathbf{t},x)(\mathbf{t}^{\prime},x^{\prime})=(\mathbf{t},x(t_{1}^{-1}t_{1}^{\prime})(Y)(x^{\prime})).$

This makes $\mathbf{P}^{a,b}(\mathcal{C})(Y)$ into an
$\mathcal{O}_{Y}(Y)$–algebra. The unity is the class of $(\mathbf{t},1)$,
indepentent of choice of $\mathbf{t}\in\Pi^{-1}_{Y}(a_{Y},b_{Y})$ by remark
4.4. From this and the fact that Cayley–Hamilton equation holds on
$\mathcal{Q}_{Y}(V)$ for any open $V\subset Y$, it follows that
$\mathbf{P}^{a,b}(\mathcal{C})(Y)$ has the same quadratic form as
$\mathcal{C}_{Y}$ [1, p.885]. So $\mathbf{P}^{a,b}(\mathcal{C})(Y)$ is an
$\mathcal{O}_{Y}(Y)$–octonion algebra.

######  Lemma 4.5.

Let $a,b\in\mathbf{S}_{\mathcal{Q_{C}}}(X)$ and
$(t_{1},t_{2},t_{3})\in\mathbf{RT}(\mathcal{C})(Y)$. Then $t_{1}(Y)$ is an
isomorphism $\mathcal{C}_{Y}(Y)\to\mathcal{C}_{Y}^{a,b}(Y)$ if and only if
$t_{2}(Y)(1)=\eta b_{Y}$ and $t_{3}(Y)(1)=\eta a_{Y}$ for some
$\eta\in\boldsymbol{\boldsymbol{\mu}}_{2}(Y)$.

###### Proof.

By definition, $\mathcal{C}_{Y}(Y)$ and $\mathcal{C}_{Y}^{a,b}(Y)$ are
$\mathcal{O}_{Y}(Y)$–octonion algebras, and $t_{1}(Y)$ an algebra homomorphism
between them. The statement then follows from [1, Proposition 4.5]. ∎

######  Lemma 4.6.

Let $a,b\in\mathbf{S}_{\mathcal{Q_{C}}}(X)$. For each $Y\in\mathbf{Sch}/X$
such that $\Pi^{-1}_{Y}(a_{Y},b_{Y})\neq\varnothing$, the map

$\displaystyle\Omega_{Y}^{a,b}:\mathbf{P}^{a,b}(\mathcal{C})(Y)$
$\displaystyle\to\mathcal{C}_{Y}^{a,b}(Y)$ $\displaystyle(\mathbf{t},x)$
$\displaystyle\mapsto t_{1}(Y)(x)$

is an algebra isomorphism.

###### Proof.

The proof is similar to that of [1, Theorem 4.6].

The map $\Omega^{a,b}_{Y}$ is well–defined for any $X$–scheme $Y$ since if
$(\mathbf{t},x)\sim(\mathbf{t}^{\prime},x^{\prime})$ then
$x^{\prime}=g(Y)^{-1}(x)$ and $\mathbf{t}^{\prime}=\mathbf{t}g$, for some
$g\in\mathbf{G}(Y)$, so

$t_{1}^{\prime}(Y)(x^{\prime})=(t_{1}g)(Y)(g(Y)^{-1}(x))=t_{1}(Y)g(Y)g(Y)^{-1}(x)=t_{1}(Y)(x).$

$\Omega^{a,b}(Y)$ is linear since,

$\Omega^{a,b}_{Y}(s(\mathbf{t},x))=\Omega^{a,b}_{Y}((\mathbf{t},sx))=t_{1}(Y)(sx)=st_{1}(Y)(x)=s\Omega^{a,b}_{Y}(\mathbf{t},x),$

for $s\in\mathcal{O}_{Y}(Y)$, and

$\displaystyle\Omega^{a,b}_{Y}((\mathbf{t},x)+(\mathbf{t}^{\prime},x^{\prime}))$
$\displaystyle=\Omega^{a,b}_{Y}(\mathbf{t},x+(t_{1}^{-1}t_{1}^{\prime})(Y)(x^{\prime}))$
$\displaystyle=t_{1}(Y)(x+(t_{1}^{-1}t_{1}^{\prime})(Y)(x^{\prime}))$
$\displaystyle=t_{1}(Y)(x)+t_{1}(Y)((t_{1}^{-1}t_{1}^{\prime})(Y)(x^{\prime}))$
$\displaystyle=t_{1}(Y)(x)+t_{1}^{\prime}(Y)(x^{\prime})$
$\displaystyle=\Omega^{a,b}_{Y}(\mathbf{t},x)+\Omega^{a,b}_{Y}(\mathbf{t}^{\prime},x^{\prime}).$

We have

$\Omega^{a,b}_{Y}((\mathbf{t},x)(\mathbf{t}^{\prime},x^{\prime}))=\Omega^{a,b}_{Y}(\mathbf{t},x(t_{1}^{-1}t_{1}^{\prime})(Y)(x^{\prime}))=t_{1}(Y)(x(t_{1}^{-1}t_{1}^{\prime})(Y)(x^{\prime})).$

Since $\mathbf{t}\in\Pi^{-1}_{Y}(a_{Y},b_{Y})$, it follows from lemma 4.5 that
this is equal to

$\displaystyle
t_{1}(Y)(x)\ast_{a_{Y},b_{Y}}t_{1}(Y)((t_{1}^{-1}t_{1}^{\prime})(Y)(x^{\prime}))$
$\displaystyle=t_{1}(Y)(x)\ast_{a_{Y},b_{Y}}t_{1}^{\prime}(Y)(x^{\prime})$
$\displaystyle=\Omega^{a,b}_{Y}(\mathbf{t},x)\ast_{a_{Y},b_{Y}}\Omega^{a,b}_{Y}(\mathbf{t}^{\prime},x^{\prime})$

If $\Omega^{a,b}_{Y}(\mathbf{t},x)=t_{1}(Y)(x)=0$ then $x=0$ since $t_{1}(Y)$
is invertible. Then

$\ker\Omega^{a,b}_{Y}=\\{(\mathbf{t},0)|\mathbf{t}\in\Pi^{-1}_{Y}(a_{y},b_{Y})\\}.$

By definition, for any
$(\mathbf{t},0),(\mathbf{t}^{\prime},0)\in\ker\Omega^{a,b}_{Y}$,
$\Pi_{Y}(\mathbf{t})=\Pi_{Y}(\mathbf{t}^{\prime})$. From remark 4.4, it
follows that $\mathbf{t}^{-1}\mathbf{t}^{\prime}\in i_{Y}(\mathbf{G}(Y))$.
Then

$(\mathbf{t}(\mathbf{t}^{-1}\mathbf{t}^{\prime}),(\mathbf{t}^{-1}\mathbf{t}^{\prime})(Y)(0))=(\mathbf{t}^{\prime},0)$

so $(\mathbf{t},0)\sim(\mathbf{t}^{\prime},0)$. Then $\ker\Omega^{a,b}_{Y}=0$
and $\Omega^{a,b}_{Y}$ is injective. Given $x\in\mathcal{C}^{a,b}_{Y}(Y)$ then
any $\mathbf{t}\in\Pi^{-1}_{Y}(a,b)\neq\varnothing$ satisfy
$\Omega^{a,b}_{Y}(\mathbf{t},t^{-1}_{1}(Y)(x))=x$, so $\Omega^{a,b}_{Y}$ is
surjective. ∎

Now we can show the link between twisted octonion algebras and isotopes.

######  Theorem 4.7.

Consider the algebra isomorphisms $\Omega^{a,b}_{Y}$, defined in lemma 4.6.
The induced map

$\widehat{\Omega}^{a,b}:\mathbf{E}^{a,b}\wedge^{\mathbf{G}}\mathbf{W}(\mathcal{C})\overset{\sim}{\to}\mathbf{W}(\mathcal{C}^{a,b}).$

is a natural isomorphism of fppf-sheaves of algebras.

######  Remark 4.8.

Naturality means that for any morphism $g:Z\to Y$ of $X$–schemes, the diagram

${\mathbf{P}^{a,b}(\mathcal{C})(Z)}$${\mathcal{C}^{a,b}_{Z}(Z)}$${\mathbf{P}^{a,b}(\mathcal{C})(Y)}$${\mathcal{C}^{a,b}_{Y}(Y)}$$\scriptstyle{\Omega^{a,b}_{Z}}$$\scriptstyle{\Omega^{a,b}_{Y}}$$\scriptstyle{\widetilde{g}:=\mathbf{P}^{a,b}(\mathcal{C})(g)}$$\scriptstyle{\widehat{g}:=\mathcal{C}^{a,b}_{Y}(g)}$

commutes. The maps are well-defined by remark 3.8.

###### Proof.

We first prove the isomorphism. It follows from the universal property of
sheafification that if the map $\Omega_{Y}^{a,b}$ on
$\mathbf{P}^{a,b}(\mathcal{C})(Y)$ is an algebra isomorphism then
$\widehat{\Omega}^{a,b}_{Y}$ is also an algebra isomorphism. Since
$\Omega^{a,b}$ is globally defined, it suffices to check an open cover of $Y$.
The (fppf) sheaf $\mathbf{E}^{a,b}\wedge^{\mathbf{G}}\mathbf{W}(\mathcal{C})$
is a $\mathbf{G}$–torsor so for every $X$–scheme $Y$ there exists an fppf
covering $\\{Y_{i}\to Y\\}_{i\in I}$ such that
$\mathbf{P}^{a,b}(\mathcal{C})(Y_{i})\neq\varnothing$. Hence we only need to
check $Y$ such that $\mathbf{P}^{a,b}(\mathcal{C})(Y)\neq\varnothing$. Let $Y$
be such a scheme. It follows from lemma 4.6 that $\Omega^{a,b}_{Y}$ is an
algebra isomorphism.

Now we prove naturality. It suffices to check locally. For any open affine
$U\subset X$ and any morphism $h:W\to V$ of $U$–schemes, the diagram

${\mathbf{P}^{a,b}(\mathcal{C})|_{U}(W)}$${(\mathcal{C}|_{U})^{a,b}_{W}(W)}$${\mathbf{P}^{a,b}(\mathcal{C})|_{U}(V)}$${(\mathcal{C}|_{U})^{a,b}_{V}(V)}$$\scriptstyle{\Omega^{a,b}_{W}}$$\scriptstyle{\Omega^{a,b}_{V}}$$\scriptstyle{\widetilde{h}}$$\scriptstyle{\widehat{h}}$

commutes [1, Theorem 4.6]. ∎

###  4.2 The equivalence of isotopes and isometries

Consider the site $(\mathbf{Sch}/X)_{\text{fppf}}$. If $\mathbf{G}$ is a sheaf
of groups in $(\mathbf{Sch}/X)_{\text{fppf}}$, we denote the set of
isomorphism classes of $\mathbf{G}$–torsors, called the _first cohomology_ ,
by $H^{1}_{\text{fppf}}(X,\mathbf{G})$ (see [8, §2.2]).

######  Lemma 4.9.

Let $(\mathcal{C},\mathcal{Q_{C}})$ be an octonion algebra over $X$. We have
bijections of pointed sets

$\displaystyle H^{1}_{\text{fppf}}(X,\mathbf{Aut}(\mathcal{C}))$
$\displaystyle\simeq\\{C^{\prime}|C^{\prime}\simeq\mathcal{C}\text{ fppf
locally}\\}$ $\displaystyle
H^{1}_{\text{fppf}}(X,\mathbf{O}(\mathcal{Q_{C}}))$
$\displaystyle\simeq\\{(\mathcal{M},\mathcal{Q_{M}})|\mathcal{Q_{M}}\sim\mathcal{Q_{C}}\text{
fppf locally}\\}$

with base points $\mathcal{C}$ and $(\mathcal{C},\mathcal{Q_{C}})$
respectively, which preserves the base points. (Here $C^{\prime}$ are octonion
algebras over $X$ and $(\mathcal{M},\mathcal{Q_{M}})$ are
$\mathcal{O}_{X}$–module sheaves with a quadratic form.)

###### Proof.

We have mutually inverse functions

$\displaystyle H^{1}_{\text{fppf}}(X,\mathbf{Aut}(\mathcal{C}))$
$\displaystyle\leftrightarrow\\{C^{\prime}|C^{\prime}\simeq\mathcal{C}\text{
fppf locally}\\}$ $\displaystyle E$ $\displaystyle\mapsto
E\wedge^{\mathbf{Aut}(\mathcal{C})}\mathcal{C}$
$\displaystyle\mathbf{Isom}(\mathcal{C},C^{\prime})$ $\displaystyle\mapsfrom
C^{\prime}$

where $E\wedge^{\mathbf{Aut}(\mathcal{C})}\mathcal{C}$ is the sheafification
of

$\mathbf{E}(\mathcal{C})(Y)=(E(Y)\times\mathcal{C}_{Y}(Y))/\sim$

where

$(u,x)\sim(u^{\prime},x^{\prime})\iff\exists
g\in\mathbf{Aut}(\mathcal{C})(Y):(ug,x)=(u^{\prime},g(Y)(x^{\prime})),$

and $\mathbf{Isom}(\mathcal{C},C^{\prime})$ is the set of isomorphisms from
$\mathcal{C}$ to $C$’. The other bijection is similar. ∎

Recall the scheme $\boldsymbol{\mu}_{2}$. We can embed this into
$\mathbf{RT}(\mathcal{C})$ by

$\displaystyle\boldsymbol{\mu}_{2}(Y)$
$\displaystyle\hookrightarrow\mathbf{RT}(\mathcal{C})(Y)$ $\displaystyle\eta$
$\displaystyle\mapsto(1,\eta,\eta)$

We also have an embedding into $\mathbf{S}_{\mathcal{Q_{C}}}^{2}$:

$\displaystyle\boldsymbol{\mu}_{2}(Y)$
$\displaystyle\hookrightarrow\mathbf{S}_{\mathcal{Q_{C}}}^{2}(Y)$
$\displaystyle\eta$ $\displaystyle\mapsto(\eta\cdot 1_{\mathcal{C}},\eta\cdot
1_{\mathcal{C}})$

We can now mod out $\boldsymbol{\mu}_{2}$ from the mapping $\Pi$, and get

$\Pi_{+}:\mathbf{RT}(\mathcal{C})/\boldsymbol{\mu}_{2}\to\mathbf{S}_{\mathcal{Q_{C}}}^{2}/\boldsymbol{\mu}_{2}$

Let $\\{U_{i}\\}$ be an affine cover of $X$ and
$f_{1}:\mathbf{RT}(\mathcal{C})\to\mathbf{SO}(\mathcal{Q_{C}})$ be the
projection onto the first coordinate. The map $f_{1}$ induces an
$U_{i}$–isomorphism

$\big{(}\mathbf{RT}(\mathcal{C})/\boldsymbol{\mu}_{2}\big{)}\big{|}_{U_{i}}\simeq\mathbf{RT}(\mathcal{C})|_{U_{i}}\big{/}\boldsymbol{\mu}_{2}|_{U_{i}}\overset{\sim}{\longrightarrow}\mathbf{SO}(\mathcal{Q_{C}})|_{U_{i}}$

[1, p. 893], where the first isomorphism comes from a similar argument as for
the quotient scheme in theorem 4.3. It follows that we have an $X$–isomorphism
$\mathbf{RT}(\mathcal{C})/\boldsymbol{\mu}_{2}\overset{\sim}{\longrightarrow}\mathbf{SO}(\mathcal{Q_{C}})$,
so we get a commutative diagram

${\mathbf{RT}(\mathcal{C})}$${\mathbf{SO}(\mathcal{Q_{C}})}$${\mathbf{S}_{\mathcal{Q_{C}}}^{2}}$${\mathbf{S}_{\mathcal{Q_{C}}}^{2}/\boldsymbol{\mu}_{2}}$$\scriptstyle{f_{1}}$$\scriptstyle{\Pi}$$\scriptstyle{\Pi_{+}}$$\scriptstyle{\rho}$

where $\Pi$ and $\Pi_{+}$ are $\mathbf{Aut}(\mathcal{C})$–torsors.

######  Lemma 4.10.

The diagram $($4.2$)$ is a Cartesian square.

###### Proof.

Let $\\{U_{i}\\}_{i\in I}$ be an affine open cover of $X$. For each $i\in I$,
the restricted diagram

${\mathbf{RT}(\mathcal{C})|_{U_{i}}}$${\mathbf{SO}(\mathcal{Q_{C}})|_{U_{i}}}$${\mathbf{S}_{\mathcal{Q_{C}}}^{2}|_{U_{i}}}$${(\mathbf{S}_{\mathcal{Q_{C}}}^{2}/\boldsymbol{\mu}_{2})|_{U_{i}}}$$\scriptstyle{f_{1}|_{U_{i}}}$$\scriptstyle{\Pi|_{U_{i}}}$$\scriptstyle{\Pi_{+}|_{U_{i}}}$$\scriptstyle{\rho|_{U_{i}}}$

is a Cartesian square [1, p. 893]. Let $Y\in\mathbf{Sch}/X$, and let
$Y\overset{g}{\to}\mathbf{SO}(\mathcal{Q_{C}})$ and
$Y\overset{h}{\to}\mathbf{S}_{\mathcal{Q_{C}}}^{2}$ be $X$–morphisms such that
$\Pi_{+}g=\rho h$. Then

$(\Pi_{+}|_{U_{i}})(g|_{U_{i}})=(\Pi_{+}g)|_{U_{i}}=(\rho
h)|_{U_{i}}=(\rho|_{U_{i}})(h|_{U_{i}}),$

so for all $i\in I$ there exists a morphism
$\lambda_{i}:Y|_{U_{i}}\to\mathbf{RT}(\mathcal{C})|_{U_{i}}$ such that

$g|_{U_{i}}=f_{1}|_{U_{i}}\lambda_{i}\text{ and
}h|_{U_{i}}=\Pi|_{U_{i}}\lambda_{i}.$

On the intersection $U_{i}\cap U_{j}$, both $\lambda_{i}$ and $\lambda_{j}$
make the restricted diagram commute. By the uniqueness in a Cartesian square,
we have

$\lambda_{i}|_{U_{i}\cap U_{j}}=\lambda_{j}|_{U_{i}\cap U_{j}}.$

Then there exists a $\lambda:Y\to\mathbf{RT}(\mathcal{C})$ such that
$\lambda|_{U_{i}}=\lambda_{i}$ [11, Lemma 6.33.1]. In particular, this
$\lambda$ satisfies

$g=f_{1}\lambda\text{ and }h=\Pi\lambda$

and we are done. ∎

We have two projections
$p_{i}:\mathbf{S}_{\mathcal{Q_{C}}}^{2}/\boldsymbol{\mu}_{2}\to\mathbf{S}_{\mathcal{Q_{C}}}/\boldsymbol{\mu}_{2}$,
$i=1,2$, defined by projection onto the first and second coordinate
respectively. We get two actions of $\mathbf{RT}(\mathcal{C})$ on
$\mathbf{S}_{\mathcal{Q_{C}}}$:

$\big{(}(t_{1},t_{2},t_{3}),u\big{)}\mapsto t_{3}(u),\quad\text{and
}\quad\big{(}(t_{1},t_{2},t_{3}),v\big{)}\mapsto t_{2}(v).$

This induces two actions of
$\mathbf{RT}(\mathcal{C})/\boldsymbol{\mu}_{2}\simeq\mathbf{SO}(\mathcal{Q_{C}})$
on $\mathbf{S}_{\mathcal{Q_{C}}}/\boldsymbol{\mu}_{2}$, which we denote by
$(g,x)\mapsto g\bullet_{i}x$, $i=1,2$.

######  Lemma 4.11.

Both actions of $\mathbf{SO}(\mathcal{Q_{C}})(X)$ on
$(\mathbf{S}_{\mathcal{Q_{C}}}/\boldsymbol{\mu}_{2})(X)$ are transitive.

###### Proof.

First we will prove that the orbit map
$\mathbf{SO}(\mathcal{Q_{C}})\to\mathbf{S}_{\mathcal{Q_{C}}}/\boldsymbol{\mu}_{2}$,
$g\mapsto g\bullet_{j}[1]$ admits a splitting for each $j=1,2$. Let $j=2$. We
have a map $\mathbf{S}_{\mathcal{Q_{C}}}\to\mathbf{SO}(\mathcal{Q_{C}})$
defined by

$\displaystyle\mathbf{S}_{\mathcal{Q_{C}}}(Y)$
$\displaystyle\to\mathbf{SO}(\mathcal{Q_{C}})(Y)$ $\displaystyle a$
$\displaystyle\mapsto B_{a}$

The map is $\boldsymbol{\mu}_{2}$–invariant, since for any $X$–scheme $Y$ and
any $\eta\in\boldsymbol{\mu}_{2}(Y)$

$B_{\eta a}(Y)(x)=\eta a\cdot x\cdot\eta a=\eta^{2}\cdot a\cdot x\cdot
a=a\cdot x\cdot a=B_{a}(V)(x),$

so it induces a map
$h:\mathbf{S}_{\mathcal{Q_{C}}}/\boldsymbol{\mu}_{2}\to\mathbf{SO}(\mathcal{Q_{C}})$.
Let $x\in(\mathbf{S}_{\mathcal{Q_{C}}}/\boldsymbol{\mu}_{2})(X)$ and
$\\{U_{i}\to X\\}$ be an fppf covering. Then for every $i$, $x_{U_{i}}$ lifts
to some $a_{i}\in\mathbf{S}_{\mathcal{Q_{C}}}(U_{i})$. Then
$h(X)(x)_{U_{i}}=B_{a_{i}}(U_{i})$, which lifts to
$(B_{a_{i}},R_{\overline{a_{i}}},L_{\overline{a_{i}}})=:\mathbf{t}\in\mathbf{RT}(\mathcal{C})(U_{i})$
(see example 4.2). We have

$h(X)(x)_{U_{i}}\bullet_{2}x_{U_{i}}=p_{1}(\mathbf{t}.x_{U_{i}})=R_{\overline{a_{i}}}(U_{i})[a_{i}]=[R_{\overline{a_{i}}}(U_{i})(a_{i})]=[1],$

and since it holds for each $U_{i}$ it follows that

$h(X)(x)^{-1}\bullet_{2}[1]=x.$

So $x\mapsto h(X)(x)^{-1}$ is a section of the orbit map. The case $j=1$ is
done similarly, but with the first projection $p_{1}$. Now, let
$x,y\in\mathbf{S}_{\mathcal{Q_{C}}}/\boldsymbol{\mu}_{2}$. We have
$h(X)(y)^{-1}\cdot h(X)(x)\in\mathbf{SO}(\mathcal{Q_{C}})(X)$, so

$(h(X)(y)^{-1}\cdot
h(X)(x))\bullet_{j}x=h(X)(y)^{-1}\bullet_{j}(h(X)(x)\bullet_{j}x)=h(X)(y)^{-1}\bullet_{j}[1]=y.$

Hence the actions are transitive. ∎

Now we have everything we need to state and prove our last theorem.

######  Theorem 4.12.

Let $\mathbf{G}=\mathbf{Aut}(\mathcal{C})$ and let

$\displaystyle S_{1}$
$\displaystyle=\ker\big{(}H^{1}_{\text{fppf}}(X,\mathbf{G})\to
H^{1}_{\text{fppf}}(X,\mathbf{RT}(\mathcal{C}))\big{)},$ $\displaystyle S_{2}$
$\displaystyle=\ker\big{(}H^{1}_{\text{fppf}}(X,\mathbf{G})\to
H^{1}_{\text{fppf}}(X,\mathbf{O}(\mathcal{Q}_{\mathcal{C}}))\big{)},$

where the maps are induced by

$\displaystyle\mathbf{Aut}(\mathcal{C})$
$\displaystyle\to\mathbf{RT}(\mathcal{C}),\qquad\qquad\mathbf{Aut}(\mathcal{C})\to\mathbf{O}(\mathcal{Q}_{\mathcal{C}})$
$\displaystyle\varphi$
$\displaystyle\mapsto(\varphi,\varphi,\varphi)\qquad\qquad\qquad\psi\mapsto\psi$

Then $S_{1}=S_{2}$.

######  Remark 4.13.

The maps are well–defined by remark 3.5 and theorem 4.3(1).

###### Proof of theorem 4.12.

Parts of this proof is inspired by the proof of theorem 6.6 in [1]. Notice
that $S_{1}$ classifies the octonion algebras over $X$ that are isomorphic to
$\mathcal{C}^{a,b}$. We have homomorphisms

$\displaystyle i_{1}:\mathbf{Aut}(\mathcal{C})$
$\displaystyle\to\mathbf{RT}(\mathcal{C}),\qquad\qquad\qquad
i_{2}:\mathbf{RT}(\mathcal{C})\to\mathbf{SO}(\mathcal{Q}_{\mathcal{C}})$
$\displaystyle\varphi$
$\displaystyle\mapsto(\varphi,\varphi,\varphi)\qquad\qquad\qquad(\psi_{1},\psi_{2},\psi_{3})\mapsto\psi_{1}$
$\displaystyle i_{3}:\mathbf{SO}(\mathcal{Q}_{\mathcal{C}})$
$\displaystyle\to\mathbf{O}(\mathcal{Q}_{\mathcal{C}}),\qquad\quad
i_{4}:\mathbf{Aut}(\mathcal{C})\to\mathbf{O}(\mathcal{Q}_{\mathcal{C}})$
$\displaystyle\rho$
$\displaystyle\mapsto\rho\qquad\qquad\qquad\qquad\qquad\varphi\mapsto\varphi$

We see that $i_{4}=i_{3}\circ i_{2}\circ i_{1}$. $H^{1}(X,\\_)$ is a covariant
functor so $i_{4}^{\ast}=i_{3}^{\ast}\circ i_{2}^{\ast}\circ i_{1}^{\ast}$,
where $i_{j}^{\ast}:=H^{1}_{\text{fppf}}(X,i_{j})$. From this it follows that
$S_{1}\subseteq S_{2}$.

Let $[C^{\prime}]\in S_{2}$. Then there exists $\varphi:\mathcal{C}\to
C^{\prime}$ such that $\varphi\in\mathbf{O}(\mathcal{Q_{C}})$. For any affine
connected subset $U\subset X$, either
$\varphi|_{U}\in\mathbf{SO}(\mathcal{Q_{C}})$ or
$\varphi|_{U}\in\mathbf{O}(\mathcal{Q_{C}})\setminus\mathbf{SO}(\mathcal{Q_{C}})$
[1, Theorem 6.6]. Let $X_{i}$ be the connected components of $X$, and
$\\{U_{ij}\\}$ an affine connected cover of $X_{i}$. If for some $j_{1},j_{2}$
we have $U_{ij_{1}}\cap U_{ij_{2}}\neq\varnothing$ and

$\varphi|_{U_{ij_{1}}}\in\mathbf{SO}(\mathcal{Q_{C}}),\text{ and
}\varphi|_{U_{ij_{2}}}\in\mathbf{O}(\mathcal{Q_{C}})\setminus\mathbf{SO}(\mathcal{Q_{C}}),$

then there exists an open affine $V\subset U_{ij_{1}}\cap U_{ij_{2}}$ such
that

$\varphi|_{V}\in\mathbf{SO}(\mathcal{Q_{C}}),\text{ and
}\varphi|_{V}\in\mathbf{O}(\mathcal{Q_{C}})\setminus\mathbf{SO}(\mathcal{Q_{C}}),$

a contradiction. Hence, for every $i$, either

$\varphi|_{U_{ij}}\in\mathbf{SO}(\mathcal{Q_{C}})\quad\forall j$

or

$\varphi|_{U_{ij}}\in\mathbf{O}(\mathcal{Q_{C}})\setminus\mathbf{SO}(\mathcal{Q_{C}})\quad\forall
j.$

Then there exists a morphism $\psi:\mathcal{C}\to C^{\prime}$ such that

$\psi|_{U_{ij}}=\left\\{\begin{array}[]{cc}\varphi|_{U_{ij}},&\text{if
}\varphi|_{U_{ij}}\in\mathbf{SO}(\mathcal{Q_{C}})\\\
\kappa|_{U_{ij}}\circ\varphi|_{U_{ij}},&\text{if
}\varphi|_{U_{ij}}\in\mathbf{O}(\mathcal{Q_{C}})\setminus\mathbf{SO}(\mathcal{Q_{C}})\end{array}\right.$

Then $\psi\in\mathbf{SO}(\mathcal{Q_{C}})$. There exists a bijection [8,
Proposition 2.4.3]

$\displaystyle\phi:\mathbf{SO}(\mathcal{Q_{C}})(X)\backslash(\mathbf{S}_{\mathcal{Q_{C}}}^{2}/\boldsymbol{\mu}_{2})(X)$
$\displaystyle\overset{\sim}{\longrightarrow}S_{2}$ $\displaystyle x$
$\displaystyle\longmapsto\Pi_{+}^{-1}(x)\wedge^{\mathbf{G}}\mathbf{W}(\mathcal{C}),$

where
$\mathbf{SO}(\mathcal{Q_{C}})(X)\backslash(\mathbf{S}_{\mathcal{Q_{C}}}^{2}/\boldsymbol{\mu}_{2})(X)$
is the set of orbits of the $\mathbf{SO}(\mathcal{Q_{C}})(X)$–action. Each
orbit can be represented by an element in
$(\mathbf{S}_{\mathcal{Q_{C}}}^{2}/\boldsymbol{\mu}_{2})(X)$, so let
$x\in(\mathbf{S}_{\mathcal{Q_{C}}}^{2}/\boldsymbol{\mu}_{2})(X)$ such that
$\phi(x)=[C^{\prime}]$ and let $x_{1}$ be its projection onto the first copy
of $(\mathbf{S}_{\mathcal{Q_{C}}}/\boldsymbol{\mu}_{2})(X)$. By Lemma 4.11, we
may assume
$x_{1}=[1]\in(\mathbf{S}_{\mathcal{Q_{C}}}/\boldsymbol{\mu}_{2})(X)$. Consider
the commutative diagram

${\mathbf{S}_{\mathcal{Q_{C}}}^{2}}$${\mathbf{S}_{\mathcal{Q_{C}}}}$${\mathbf{S}_{\mathcal{Q_{C}}}^{2}/\boldsymbol{\mu}_{2}}$${\mathbf{S}_{\mathcal{Q_{C}}}/\boldsymbol{\mu}_{2}}$$\scriptstyle{\varrho_{1}}$$\scriptstyle{\varrho_{2}}$

where the horizontal maps are projections onto the first coordinate. Note that
$\varrho_{1}$ and $\varrho_{2}$ define $\boldsymbol{\mu}_{2}$–torsors over
$\mathbf{S}_{\mathcal{Q_{C}}}^{2}/\boldsymbol{\mu}_{2}$ and
$\mathbf{S}_{\mathcal{Q_{C}}}/\boldsymbol{\mu}_{2}$ respectively. By the
commutativity of the diagram, we get an isomorphism
$\varrho_{1}^{-1}(x)\to\varrho_{2}^{-1}([1])$ of
$\boldsymbol{\mu}_{2}$–torsors. Since
$\varrho_{2}^{-1}(X)([1])=\boldsymbol{\mu}_{2}(X)\cdot 1\neq\varnothing$, we
have $\varrho_{1}^{-1}(x)\neq\varnothing$, so $x$ lifts to an element
$(a,b)\in\mathbf{S}_{\mathcal{Q_{C}}}^{2}(X)$. By the Cartesian diagram
$($4.2$)$ we have a morphism
$\Pi^{-1}(a,b)\overset{f_{1}}{\to}\Pi_{+}^{-1}(x)$ and, since $\Pi$ and
$\Pi_{+}$ are $\mathbf{G}$–torsors, this morphism must be an isomorphism. Then
we have an isomorphism of fppf–sheaves of algebras

$\mathbf{W}(\mathcal{C}^{a,b})\simeq\Pi^{-1}(a,b)\wedge^{\mathbf{G}}\mathbf{W}(\mathcal{C})\simeq\Pi_{+}^{-1}(x)\wedge^{\mathbf{G}}\mathbf{W}(\mathcal{C})\simeq\mathbf{W}(C^{\prime}).$

Thus $[C^{\prime}]\in S_{1}$. ∎

######  Corollary 4.14.

Let $\mathcal{C}$ and $\mathcal{C}^{\prime}$ be two octonion algebras over X.
The quadratic forms $\mathcal{Q}_{\mathcal{C}}$ and
$\mathcal{Q}_{\mathcal{C}^{\prime}}$ are isometric if and only if there exists
$a,b\in\mathbf{S}_{\mathcal{Q_{C}}}(X)$ such that $\mathcal{C}^{\prime}$ is
isomorphic to $\mathcal{C}^{a,b}$.

###### Proof.

By theorem 4.7 and lemma 4.9, the set $S_{1}$ classifies the octonion algebras
over $X$ isomorphic to $\mathcal{C}^{a,b}$ for some
$a,b\in\mathbf{S}_{\mathcal{Q_{C}}}(X)$. Also by lemma 4.9, the set $S_{2}$
classifies octonion algebras whose norm is isometric to $\mathcal{Q_{C}}$. By
the equality established in theorem 4.12, the statement follows. ∎

## References

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# Guessing human intentions to avoid dangerous situations in caregiving robots

Noé Zapata1, Gerardo Pérez1, Lucas Bonilla1, Pedro Núñez1, Pilar Bachiller1
and Pablo Bustos1 1All authors are members of RoboLab, University of
Extremadura, Spain, (nzapatac, gperezgonz, lbonillar, pnuntru, pilarb,
<EMAIL_ADDRESS>work has been partially funded by the Spanish Ministry of
Science and Innovation TED2021-131739-C22, supported by
MCIN/AEI/10.13039/501100011033 and the European Union NextGeneration EU/PRTR,
by the Spanish Ministry of Science and Innovation PDC2022-133597-C41, by FEDER
Project 0124_EUROAGE_MAS_4_E (2021-2027 POCTEP Program) and by the Spanish
Ministry of Science and Innovation PID2022-137344OB-C31 funded by
MCIN/AEI/10.13039/501100011033/FEDER, UE.

###### Abstract

For robots to interact socially, they must interpret human intentions and
anticipate their potential outcomes accurately. This is particularly important
for social robots designed for human care, which may face potentially
dangerous situations for people, such as unseen obstacles in their way, that
should be avoided. This paper explores the Artificial Theory of Mind (ATM)
approach to inferring and interpreting human intentions. We propose an
algorithm that detects risky situations for humans, selecting a robot action
that removes the danger in real time. We use the simulation-based approach to
ATM and adopt the ”like-me” policy to assign intentions and actions to people.
Using this strategy, the robot can detect and act with a high rate of success
under time-constrained situations. The algorithm has been implemented as part
of an existing robotics cognitive architecture and tested in simulation
scenarios. Three experiments have been conducted to test the implementation’s
robustness, precision and real-time response, including a simulated scenario,
a human-in-the-loop hybrid configuration and a real-world scenario.

##  I Introduction

Caregiving robots can be seen as special social robots that, among other
things, react whenever they perceive a dangerous situation for a human. To
enable smooth human-robot collaboration, these robots must be aware of their
social environment by showing advanced levels of reactivity, interaction, and
comprehension [1]. Cultural norms, social cues, human activities, and
individual preferences of people around them typically influence social
awareness in robots. To be socially adept, robots must perceive their
surroundings and occupants, comprehending the situation’s social and
contextual nuances.

Figure 1: Schematic view of the model. The robot assigns two targets to the
human, the door and the couch, and computes one trajectory in which the person
collides with the ball. To save the situation, the robot imagines several
possible actions to check if one removes imminent danger. The smaller frame
shows the scene rendered by the simulator.

Caregiving robots, in particular, must be able to sense and interpret the
ongoing activities of individuals in their environment to anticipate future
risk scenarios and adjust their behaviour to mitigate the situation in a
socially acceptable manner. To achieve this, robots must be able to predict
forthcoming events, foresee potential consequences, and select the most
understandable, socially acceptable behaviour to align with human social
expectations.

In this paper, we build on previous results in the field of artificial theory
of mind (ATM) [2] and its relation to simulation-based internal models [3][4].
Theory of mind (TM) refers to the ability to predict the actions of oneself
and others [5]. It provides a plausible explanation of how we can anticipate
the behaviour of others in particular circumstances [3][6]. Within TM, there
are several approaches to explain how the underlying processes work. We follow
the simulation theory (ST) variant here, which states that ”… people use
imagination, mental pretence, or perspective taking (’putting oneself in the
other person’s shoes’) to determine the mental states of others” [7]. Our
contribution here is to support further the hypothesis that simulation-based
internal models can provide a feasible basis for ATM and, on this basis, to
provide a working model and an example of human caregiving in social robots.

We propose a model that assigns intentions to humans using the same machinery
used by our robot to attach itself to local goals. This approach is known as a
”like-me” simulation [6]. As part of its existing set of capabilities, our
robot Shadow[8] can select a visually detected object as a target and navigate
towards it along an obstacle-free path. Accordingly, nearby humans are
assigned intentions as links between them and the objects in their
environment. The robot interprets these links as the potential to perform an
action involving the person and the object, where possible actions are those
the robot can already do. To understand the consequences of human actions, the
robot simulates them using its internal model but adopting a third-person
perspective. Following [9], we assume that the robot understands a phenomenon
if it has a model for it and the logical implications in the model correspond
to causality in the system. To explore this approach further, we have provided
the robot with a simple model of itself and its interactions with the
environment. This model can be run by an internal physics-based simulator and
used to explore the simulated consequences of its actions and derive a plan to
execute. Following this line of reasoning, an interesting way of applying this
theory of understanding to the intentions that our model assigns to people is
to simulate their possible actions and check whether they have consequences in
the environment and whether they can be a security risk. If an intention
results in a risk to the person, the robot will continue to simulate whether
any of its actions would eliminate the risk, and if so, it will carry them
out. This idea is shown in Figure 1, where a person is assigned two targets,
the door and the table, each giving rise to several trajectories. As one of
them is considered dangerous, the robot imagines an action that could revert
the situation. The way to confirm that the action will be effective is to
simulate the person’s trajectory again but with the robot placed at its
imagined final destination. As a result, the person acknowledges the presence
of the larger volume added by the robot and plans an obstacle-free trajectory.

The paper proceeds by describing other works in ATMs that have been applied to
social robots. Our model is then described in detail, followed by the most
relevant aspects of its implementation in the CORTEX cognitive architecture
[10][11]. To conclude, we will show the results of the simulated and real-
world experiments and provide some closing remarks.

##  II Related work

A pioneering work in Artificial Theory of Mind was the work by Scassellatti
[2], where a detailed framework is proposed and implemented in the humanoid
robot Cog, triggering the research in this field. An early work that uses
simulation to model ATM is [6]. Their authors conduct the experiments in the
embodied version of ACT-R/E. One of the experiments shows that the ”like-me”
approach can be used to determine where the person is looking when the robot’s
sensors are not accurate enough. However, they do not attempt to model it as
an intention nor to transform it into a plan that can be simulated and
evaluated in the model. Gray et al. [12] describe an ATM experiment in which
the robot tries to manipulate the human’s beliefs by concatenating two
primitives: action-simulation and mental-state-simulation. These states
correspond with our simulation of human actions before and after the robot
intervention. Although the paper presents a sophisticated scheme for
recursively modelling beliefs, its goal is to show how embodiment connects
agents’ mental states rather than applying ATM to a social robot in real-time.

A more recent series of works by A. Winfield and collaborators [3][4] have
shown an architecture for a robot with a simulation-based internal model named
consequence engine. Although their and our approaches are initially similar,
their robots live in a 2D grid. Their simulated actions are projections of
their state into the future using the existing movement controller. In our
model, people are assigned intentions as potential relations with visually
detectable objects, which are then transformed into actions. As a result, the
robot operates in a more abstract representation akin to adaptation and
learning. Using a different approach, Lemaignan et al. [13] propose an
architecture with an ATM module in which the simulation is carried out by a
high-level planner using shared plans. In their planning domain, agents
(humans) are first-class entities. This approach extends ATM into the Theory
Theory (TT) variation that differs from using an internal model-based
simulator [14]. Some other works have explored the relationship between the
use of ATM in robots and its effect on people’s acceptance or trust [15], or
its connection with ethics and security [16].

From a broader perspective, detecting human intentions has gained interest in
social robotics and, particularly, in socially aware robot navigation [17].
For instance, in [18], the robot accompanies a person while predicting their
intentions to reach a destination point. Human intentions regarding their goal
positions are also detected and used in [19]. Their proposal considers places
with semantic meaning as potential human goal positions. In [20], the proposed
system detects signals of intentions or preferences over avoidance strategies
and makes the agent act towards simplifying people’s decision-making. The work
in [21] presents a control scheme for an intelligent wheelchair navigation
assistant that detects cooperative and non-cooperative behaviours by comparing
the predicted pedestrian positions to the real ones. We concur with these
works in characterising intentions, but our experiment also explores how
intentions are converted into actions and what their consequences may be.
Another interesting approach, more in line with the one presented in this
work, is proposed in [22]. Given samples from an agent policy, this approach
simulates the robot and the agents forward under their assigned policies to
obtain predicted states and observation sequences. The robot aims to find an
optimal policy based on these observations over a given decision horizon.
Their work differs from ours in how the context is represented and how the new
goal, i.e. dangerous situations, is detected and integrated into the search
for a successful action.

##  III The scenario

As a test bed for our research, we have created a scenario using the
Webots111https://cyberbotics.com/ simulator. As shown in Figure 1, a robot
observes a person that is about to cross the room, heading towards a couch on
the opposite wall. A small object is on the floor, just on the way to the
target. The robot has to guess the person’s intentions, how they will unfold
and, in case of a possible collision, how to act to avoid it.

The robot is a custom-built unit named Shadow [8], that has a rectangular base
with four Mecanum wheels. The robot’s main sensors are a 360º camera placed on
top of a 3D LiDAR. Both devices are situated in the uppermost part and are co-
calibrated. The 3D pointcloud is projected onto the camera image to provide a
sparse depth plane that is used to estimate the distance to regions in the
image. All objects in the scene are recognisable by the You Only See Once
(YOLO)222We use YoloV8 from Ultralytics. https://www.ultralytics.com/ deep
neural network [23]. There are no other distractors in the room. If the
detected element is a person, its skeleton is passed to a second DNN,
JointBDOE333https://github.com/hnuzhy/jointbdoe, to estimate their orientation
[24]. All objects detected are assigned a depth coordinate obtained from the
LiDAR. Also, they are tracked using the ByteTrack algorithm [25] to assign
them a persistent identification tag. We define a visual element $e$ as
belonging to the set of YOLO recognisable objects $VE$. The stream of detected
visual elements is defined as $e_{t}\in\\{VE\\}$, where $t\geq 0$. The robot
in this scenario can only execute one action, move_to($e_{i}$), where $e_{i}$
is a visually detectable object in the scene. It does not have a speaker. The
action is executed by a local controller consisting of several components: a
path planner over a local grid, a model predictive control over the generated
trajectory, and a virtual bumper [8]. No global map is needed to execute
visually guided motions. The robot has a known maximum speed and acceleration.
A new target $e\in\\{VE\\}$ is sent to the local controller whenever the robot
is required to approach a different object in the scene.

##  IV The architecture

We use for the experiments a simplified version of the CORTEX robotics
cognitive architecture [10]. Figure 2 shows a general outline and the elements
used in this research.

Figure 2: The CORTEX architecture.

CORTEX is a multi-agent architecture designed to facilitate the creation of
information flows among different types of memories and modules. These flows
are pumped by agents. Agents are represented as semicircular forms connected
to the working memory $\mathcal{W}$. They can edit it to create and maintain
an updated context accessible to the rest of the agents. $\mathcal{W}$ is a
distributed graph structure implemented using conflict-free replicated data
types (CRDT) [26]. Its nodes are elements of a predefined ontology, and the
edges can encode geometric transformation or logic predicates. Both nodes and
edges can store a list of attributes of a predefined type.

The key elements needed to develop this experiment are a) the working memory
$\mathcal{W}$; b) the block labelled sub-cognitive level that consists of
several pipelines of components providing $\mathcal{W}$ with a stream of
visually detected elements, and also holds the navigation stack which drives
the robot to a visual target; c) the agent connecting the sub-cognitive module
with $\mathcal{W}$; d) the internal simulator, based on
PyBullet444https://pybullet.org/ and the agent that interfaces it; and e) the
two new agents that implement the ATM and that are described in the next
section. More details on the implementation of the working memory can be found
in [27].

##  V The model

In the ”like-me” interpretation of the simulation-based ATM, the robot uses
its internal model to attribute intentions to people and to simulate their
behaviour, i.e. the robot reads people as robots. We define an intention in
this scenario as the engagement that a person establishes with an object in
the environment, possibly through its affordances. Intentions are enacted by
linking them to one or more of the robot’s potential actions and then
simulated with its internal model. The effects are believed to be causal in
the world and can give rise to real actions. The context $ctx$ is defined as
the content of the working memory $\mathcal{W}$, including all the current
robot beliefs about its environment. The internal simulator typically runs
synchronously with the context, effectively maintaining a copy of the state.
When the ”like-me” process starts, the simulator is temporarily detached, and
the people’s intentions and the robot’s actions can be safely simulated. Upon
completion, the simulator goes back to synchronisation mode.

We have implemented two algorithms that read and modify the working memory,
$\mathcal{W}$. The first agent is activated when people are in $\mathcal{W}$.
It will guess and enact their intentions, marking some of them as dangerous if
a collision is detected. The marked intentions activate the second agent,
seeking a feasible robot’s intervention that cancels it. Both agents
communicate through annotations in $\mathcal{W}$. Two predefined lists are
provided to simplify the notation: $Gaze$ and $Actions$, which contain the
minimum and maximum values for the vertical gaze of a generic human and the
list of the robot’s actions, respectively.

Algorithm 1 intention guessing and enacting

1:$W,Gaze,Actions$

2:$W$

3:$ctx\leftarrow getCotext(W)$

4:$People\leftarrow getPeople(ctx)$

5:$Objects\leftarrow getObjects(ctx)$

6:for all $p_{i}\in{People}$ do

7: $Tgts\leftarrow fov(p_{i},Objects)$

8: for all $t_{j}\in Tgts$ do

9: for all $act_{k}\in Actions$ do

10: for all $gaze_{l}\in sample(Gaze)$ do

11: $int_{ijkl}\leftarrow intention(p_{i},t_{j},act_{k},gaze_{l})$

12: $c\leftarrow sim(ctx,int_{ijkl})$

13: $W\leftarrow addToW(W,(int_{ijkl},c))$

14: end for

15: end for

16: end for

17:end for

Algorithm 1 starts by accessing the current context $ctx$ provided by the
working memory $\mathcal{W}$ and the lists of people and objects of the
robot’s environment. For each person, the list of objects entering their field
of view ($Tgts$) is obtained using a simple geometric inclusion test over a
predefined frustum (line 5). Each object is assumed to be a potential
interaction target for the person. Thus, for each person and target,
intentions enacted by potential actions must be simulated to predict risky
situations. An intention is considered dangerous if its enactment entails a
collision. Since the risk assessment depends on the person’s ability to
perceive the situation, the person’s gaze is also considered during the
simulation. We assume the robot cannot estimate it from its position, so the
predefined range in $Gaze$ is uniformly sampled (line 8). According to the
person ($p_{i}$), the object ($t_{j}$), the action ($act_{k}$), and the gaze
($gaze_{l}$), an intention $int_{ijkl}$ is finally generated (line 9). The
simulator is then called to execute that intention given the current context
($ctx$) (line 10). This call implies freezing the current context so changes
can be made without interfering with the perception of reality. The simulation
proceeds by executing action $act_{k}$ under the constrained access to objects
in the scene given by $gaze_{l}$. In this experiment, the robot’s only action
is goto(x), so the simulator proceeds in two steps: a) a path-planning action
to compute a safe route from the person to the target object, in which the
occupancy grid is modified to include only the objects in the person’s field
of view; and b) the displacement of the person along that path in the
unmodified scene. The first step corresponds to the question ”how does the
person move through her environment given the assigned gaze?” and the second
to ”how does the robot imagine from its point of view what the person will
do?” The path is executed using a copy of the robot’s path-following
controller. The simulation provides the answers to the two questions,
returning a flag $c$ signalling the occurrence of a collision. Finally, in
line 11, the working memory is updated to include the new intention with the
attribute $c$, possibly marking a risky situation.

Algorithm 2 Action selection

1:$W$, $Actions$

2:$W$

3:$ctx\leftarrow getContext(W)$

4:$Intentions\leftarrow getIntentions(ctx)$

5:$Objects\leftarrow getObjects(ctx)$

6:for all $int_{ijkl}\in Intentions$ where $int_{ijkl}\in{Risk}$ do

7: for all $act_{m}\in Actions$ do

8: for all $o_{n}\in Objects$ do

9: $robot_{mn}\leftarrow intention(robot,o_{n},act_{m})$

10: $c\leftarrow sim(ctx,[robot_{mn},int_{ijkl}])$

11: if $\neg c$ then

12: $W\leftarrow addToW(W,robot_{mn})$

13: end if

14: end for

15: end for

16:end for

Algorithm 2 runs in a different agent and initiates when an intention is
marked as risky in $\mathcal{W}$. The algorithm iterates over the robot’s
action set, $Actions$, to check if executing any of them removes the risky
intention. There is no a priori knowledge of which action to try first or if
any of them will succeed. The first loop goes through the intentions marked as
dangerous (line 4). Then, following a similar procedure as in Algorithm 1, the
robot assigns to itself intentions to all visible objects. Each robot’s
intention $robot_{nm}$ along with the person’s intention $int_{ijkl}$ are sent
for simulation (line 8). This call entails the re-enaction of $int_{ijkl}$
but, this time, also considering the robot’s action. If that action makes the
collision disappear, it is selected for execution in the real world by adding
the corresponding robot’s intention to the working memory.

##  VI Experiments

The two proposed algorithms have been implemented as new agents in the
architecture and tested with three experiments. The first one consists of a
series of simulations on Webots where some free parameters are sampled. The
second one introduces uninformed human subjects in the loop to evaluate their
reactions to the robot intervention, and the third experiment is performed
with a real robot and person.

These experiments are designed to test the ATM model and its implementation as
a means of detecting potentially dangerous situations caused by people’s
intentions. We assume that people always choose the risky path. Thus, we do
not try to predict and detect other possible choices.

The first experiment consists of 180 initialisations of the nominal scenario,
letting the robot analyse the situation and decide which action to take. The
sampled variables are the positions of the robot, the ball, and the person.
The range of positions used in all cases is a $1.5m$ radius. The human always
follows a direct trajectory towards the couch at a constant speed of $0.5m/s$.
Figure 3 shows the different elements involved in the experiment. The graph in
the upper-left shows the current state of $\mathcal{W}$ with the node labelled
Shadow at the centre. The perceived objects and the person are connected to
the robot with RT edges that store an $SE(2)$ geometric transformation. The
rest of the edges are 2-place logic predicates representing perceived
relationships between nodes. In this example, a has_intention edge connects
the person to their intended destination, and a collision node signals that
they may implement their intention via an unsafe path. At the top right, the
contents of $\mathcal{W}$ are graphically displayed. The person is represented
by a yellow circle connected to a green cone, which limits their field of
view. Two paths connect the person to the couch. One of the paths crosses the
obstacle, represented as a red box, indicating a possible collision. The couch
is drawn as a green box. The robot is depicted twice, in red occupying its
starting position and in light green occupying the imagined position that will
make the person change their path. The lower left image is a zenithal view of
the scene, and the right image shows the PyBullet scene used to simulate the
robot and the person traversing the paths.

Table I shows the result of 180 runs with the Webots simulator. Of the total
number of experiments, $13$ was discarded due to the absence of detected
intentions. Considering the remaining experiments ($167$), the proposed
intention-guessing algorithm provides an accuracy rate of $79.64\%$. The
primary source of loss in accuracy is the detection of false positives. This
means that the robot wrongly detects some risky situations that would, in
reality, not pose a danger to the person. The primary causes of these false
positives are the inaccurate sizes of the shapes representing the person and
the obstacles in the internal simulator, and the errors in the estimations of
real distances due to noise in the LiDAR measurements. Nevertheless, the
algorithm does not produce false negatives, i.e., all the unsafe intentions
are correctly identified by the robot and no risky situation is left
unattended. Both facts can also be derived from the resulting precision and
recall values. Given the consequences of failing to detect jeopardised
intentions, recall can be considered more vital than precision. Taking this
into consideration, we measured F1 and F2 scores [28]. While the obtained F1
score is moderately high, reaching a value of approximately 0.76, the F2 score
indicates a high level of effectiveness.

Figure 3: Combined view showing the contents of $\mathcal{W}$ (up-left); a
graphical representation of $\mathcal{W}$ with two paths going from the person
to the couch (up-right); a zenithal view of the scene as rendered by Webots
(down-left); and a 3D view of the internal simulator, PyBullet, with simple
geometric forms representing the elements in the scene (down-right)

Regarding time metrics, the mean reaction time, which is the period going from
detecting a possible collision to the appearance of an action that can avoid
it, is less than $0.75s$. The experiment suggests that a under one second
response time is enough for real-time operation in this scenario.
Additionally, the robot can generate an action that prevents the person from
colliding in most situations detected as risky. Although reaction time will
grow with the number of objects in the scene, people, and actions with their
corresponding free parameters, given some heuristics and pruning strategies
discussed in the next section, we believe it can still be kept low enough to
work in more complex scenarios.

TABLE I: Summary of Experimental Results Metric | Value
---|---
Total Experiments | 180
Experiments Discarded | 13
Valid Experiments | 167
Accuracy Rate | 79.64%
False Positives | 20.35%
False Negatives | 0%
Precision | 0.61
Recall | 1.0
F-1 Score | 0.76
F-2 Score | 0.89
Reaction time (mean) | 747ms
Reaction time (deviation) | 310ms
Experiments with Detected Collision and Action Generated | 85.18%

A second experiment was conducted to test the reaction of human subjects
unfamiliar with the problem to the robot’s intervention. This experiment has a
human-in-the-loop configuration, where six subjects were instructed to control
the person’s trajectory in the simulator using a joystick and a view from a
virtual camera placed on the avatar. The subjective camera is oriented in a
way that prevents the subjects from seeing the ball on the floor. The
subject’s goal is to reach the couch on the other side of the room, and they
are only allowed to play with the joystick for a few seconds. After completing
the experiment, all subjects avoided the hidden ball when the robot entered
their field of view and was detected. Qualitatively, all subjects generated
different trajectories with unequal free margins to the unseen obstacle but
completed the task without incidences. These variations were not considered
relevant to the experiment. Figure 4 shows a sequence of six frames from one
of these trials. The series runs from top left to bottom right, and each frame
is split into two: the Webots zenithal view of the scene at the top and the
subjective view shown to the human subject at the bottom. In the sequence, as
the robot enters the field of view, the person corrects the trajectory and
drives the avatar to the target position.

Figure 4: Human-in-the-loop experiment (top left to bottom right). The upper
half of each frame is the zenithal view rendered by the Webots simulator. The
red and blue axes mark the centre of the scene. The lower half is the view
shown to the subject where the ball is missing. When the robot approaches the
obstacle, frames 2-3, the subject turns left, overcoming it and safely
reaching the couch. Figure 5: Real world experiment (left to right). The upper
half of the frame shows the view from the robot’s camera. The lower half shows
a schematic view of the working memory with the person represented as a yellow
circle, the backpack on the floor as a red square and the target chair as a
green square. The robot is coloured dark red. The subject walks distractedly
towards the chair (frame 1) and reacts when the robot starts moving (frame 2),
changing direction and continuing.

The third experiment has been performed with the Shadow robot [8] in a real
scenario. A group of five subjects were instructed to cross the room while
reading a paper and heading towards a chair on the other side. A backpack is
in the way555We replaced the football with a backpack because, in the real
scenario, YOLO detected it much more confidently., and they are instructed to
ignore it. They were not briefed about the robot participation. In all five
trials, the robot detected the subject and moved close to the unseen obstacle
to prevent the person from stumbling. The subjects reacted to the robot’s
intervention with varying degrees of surprise due to the unexpectedness of the
event, but all continued on their way to the assigned target. This variety of
people’s reactions could be used to adapt the internal model according to a
comfort metric, resulting in smoother, less startling movements. This idea is
commented on in the next section. Figure 5 shows a three-frame sequence of one
of the trials. The lower part of each frame shows a graphic representation of
the working memory with the robot in brown heading upwards, the person in
yellow heading downwards, the backpack in red, and the chair in green. The
line of yellow dots shows the path attributed to the subject during the
internal simulation phase.

##  VII Discussion and Conclusions

ATM is slowly gaining momentum as a tool for adding third-view reasoning
capabilities to robotics control architecture, but there are still many open
issues. Our research shows that an internal physics-based simulator embedded
in the robot’s control architecture could be the key to integrating ATM into
the set of cognitive abilities. The current technology allows direct and
precise control of the simulator’s scene graph and time step. However, this
integration relies on the existence of a working memory offering a stable and
updated view of the current context based on a hybrid numeric-symbolic
representation. This memory relies, in turn, on other modules which provide
the necessary connections to the world. Putting all these together implies a
considerable development effort and, as we see it, the perspective to further
pursue this task is linked to three key aspects related to the feasibility of
the approach: a) the stability and reliability of the working memory
representing the current context in which inferences can be made; b) the
combinatorial explosion caused by the nested relationships between
participants and the free parameters of their actions; and c) the robustness
required for reliable real-time execution in more open scenarios.

The multi-agent CORTEX architecture addresses the first issue to a reasonable
amount, with the working memory $\mathcal{W}$ being updated by agents that
connect to the subcognitive modules. Although there is ample room for
improvement, the results show that the representation is stable and can handle
a real-world, real-time situation under fairly realistic conditions. In
addition, the underlying data structure, which holds symbolic and numeric
values, can be monitored to provide a clear view of the internal reasoning
process, facilitating the path to accountability and ethics.

A second problem is the exponential growth caused by the nested searches that
unfold all possible relationships among the robot, its actions, the objects in
the scene and the people. This search can be curtailed if we consider ways to
exit the loops as soon as possible or even to skip them altogether. Studies in
humans suggest that no more than two or three alternative strategies are
evaluated, at least without interrupting the ongoing behaviour and resorting
to more extended reflection [29][30]. A first prune can be achieved by forcing
time and space constraints. In our experiments, the person-to-collision time
limits the admissibility of a robot’s action, i.e. displacements that will not
arrive in time are discarded. Another limitation comes from the maximum number
of people that can be represented in $\mathcal{W}$ simultaneously, which is an
architecture parameter. This number could be further reduced by distance to
the robot, relative position or some person’s attributes that could be
relevant in the current context. A further reduction can be achieved if the
search ends as soon as an action is found that removes the danger, but more
efficiently if the sets are ordered according to some metric or drawn from an
a priori distribution learnt from experience. For example, intentions could be
assigned to objects according to prior knowledge of the person’s habits, and
actions could be assigned to intentions biased by the objects’ affordances.

The third aspect, robustness, is particularly relevant for robot operation in
human-populated scenarios. Robustness depends on the stability of the
representation, on the ability to learn from experience the optimal values of
actions and objects free parameters, i.e. robot speed, human habits, specific
place to stop, time to start, etc., in each context, and on the ability to
recover from failures and unforeseen events during the execution. This
suggests that to improve the robustness, the control architecture has to be
more adaptable at each level of its organisation, integrating learning
algorithms that capture the effect of the interventions on humans in specific
contexts to fine-tune the free parameters.

An interesting issue arises when more than one person is on the scene, and
they have assigned risk intentions. In this case, the robot would face the
dilemma of saving one and ignoring the other based. This kind of decision
would involve a more complex evaluation of the situation, using a system of
values and thus entering the emerging field of social robot ethics [31].

The work presented in this paper is a new step towards an ATM that can be
effectively run inside a robotics cognitive architecture. We rely on a working
memory that holds a representation stable enough for an inner simulator to
take third-view perspectives. In future work, we expect to increase the number
and complexity of the robot’s actions while keeping the number of combinations
small, using algorithms that learn an optimal selection order from experience,
anytime design techniques or sorting heuristics from external knowledge
sources.

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