Colour Algebras over Rings

Abstract

Colour algebras are noncommutative Jordan algebras and closely related to octonion algebras. They initially emerged in physics since the multiplication of a colour algebra over the real or complex numbers describes the colour symmetry of the Gell–Mann quark model. Over fields of characteristic not equal to two, their structure is now well-known. We initiate the study of colour algebras over a unital commutative base ring R where two is an invertible element, and show when colour algebras can be constructed canonically by employing nondegenerate ternary hermitian forms with trivial determinant. We investigate their structure, their automorphism group and their derivations. We show that there is again a close connection between the colour algebras obtained from hermitian forms and certain types of octonion algebras.

1. Introduction

When studying algebras over rings instead of fields, their structure emerges more clearly, as many structures “collapse” and become invisible over base fields. For instance, the structure of octonion algebras over rings is much more interesting than the one of octonion algebras over fields, and the structure of Azumaya algebras is more exciting than the structure of central simple algebras.

The same is true for the structure of colour algebras over rings, also because of their intricate connections with octonion algebras, and for the vector products associated with both algebras. Colour algebras have, however, so far only been studied over fields.

Colour algebras over fields F of characteristic not equal to two were investigated extensively [1,2,3,4,5]. Every colour algebra over a field is flexible and quadratic, and therefore a noncommutative Jordan algebra. Indeed, there are only two nontrivial cases in the classification of finite-dimensional central simple noncommutative Jordan algebras over a field of characteristic not equal to two [6]. Colour algebras constitute one of these two nontrivial cases.

We will initiate the study of colour algebras over unital commutative base rings R where two is invertible without aiming for completeness. Our focus will be on those colour algebras that we can construct by employing nondegenerate hermitian spaces of rank three. Our initial results show that the flavour of the theory again becomes richer over base rings and will be worth more detailed investigations in a future paper.

There are three obvious possible ways to define a colour algebra over a ring; “globally” as was done for composition algebras over rings in [7], “locally” as Azumaya algebras are sometimes defined in the literature, especially when studied over schemes, or via a set of defining identities. We decided on the second approach and will use the following definition.

Let A be an algebra over R which is a finitely generated projective of constant rank as an R-module and has full support. For every prime ideal P of R, let $k\left(P\right)=R_P/m_P$ be the residue class field, where $m_P$ is the maximal ideal of the localisation $R_P$ of the ring R at P. If for all prime ideals P in R, the localised algebra $A\left(P\right)=A\otimes k\left(P\right)$ is a colour algebra over the residue class field $k\left(P\right)$, then A is called a colour algebra over R.

The “classical” split colour algebra $\mathrm{Col}\left(R\right)$ is defined on a free R-module of rank seven by employing the same multiplication as the multiplication that defines a split colour algebra over a base field. So this is an example of a “global” definition of an algebra. However, this is not the only colour algebra that splits locally, i.e., is a split colour algebra at all localisations of R. For projective R-modules T of constant rank three such that ${\bigwedge }^3\left(T\right)\cong R$ via some isomorphism $\alpha$, we define split colour algebras $\mathrm{Col}(T,\alpha )$ with underlying R-module structure $R\oplus T\oplus \check{T}$ that canonically generalize $\mathrm{Col}\left(R\right)$. They form an important class of colour algebras, and are introduced in Section 3.1 after the basic definitions are collected in Section 2. They are closely related to Zorn algebras (i.e., the split octonion algebras) over R. The construction of $\mathrm{Col}(T,\alpha )$ is functorial in both parameters involved (Proposition 1).

In Section 3.2, we construct colour algebras employing nondegenerate ternary hermitian forms with a trivial determinant. It is not clear if every colour algebra C over R can be constructed this way out of a hermitian form, however. While this is the case locally, i.e., for all $C\left(P\right)=C\otimes k\left(P\right)$ over the field $k\left(P\right)$ for all $P\in \mathrm{Spec}\left(R\right)$, for a general colour algebra C over R with R-module decomposition $C=R\oplus P$, P need not be an S-module for some suitable S.

One of our main results shows that a close (nonlocal) connection between colour algebras and octonion algebras over base rings can be established, as long as the algebras have a quadratic étale subalgebra S. Over base fields of characteristic not equal to two, this result was proved in ([3] Theorem 3.1). In the field setting, this establishes a stronger result than for algebras over rings, however, as it is not clear at all if all colour algebras over rings can be constructed employing hermitian forms.

Theorem 1.

(Theorem 2) Let S be a quadratic étale algebra over R. There are one-to-one correspondences between the following sets:

(i) 

The set of all pairs $\left(\mathrm{Cay}\right(S,P,h,\alpha ),S)$ of isomorphism classes of octonion algebras over R and quadratic étale algebras S.

(ii) 

The set of all pairs $\left(\mathrm{Col}\right(S,P,h,\alpha ),S)$ of isomorphism classes of colour algebras over R and quadratic étale algebras S.

(iii) 

The set of all pairs $\left(W\right(S,P,h,\alpha ),S)$ of isomorphism classes of vector algebras over R and quadratic étale algebras S.

In Section 4, we generalize results on isomorphisms, automorphisms, and derivations from ref. [3] to colour algebras over rings.

In the second part of our paper, we turn to colour algebras over schemes, which can be defined in the obvious way. This allows us to construct explicit examples of non-split colour algebras with non-free underlying module structures over curves of genus one in Section 6, and to systematically construct, for all choices of $l,m\in \mathbb{Z}$, noncommutative quadratic Jordan subalgebras of the split colour algebra $\mathrm{Col}\left(R[t_0,\dots ,t_n]\right)$ that are free R-modules of rank $1+\left({\displaystyle \genfrac{}{}{0pt}{}{l+n}{n}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{m+n}{n}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{(l+m)+n}{n}}\right)$ in Section 5.

For the latter, we use split colour algebras over the n-dimensional projective space ${\mathbb{P}}_R^n$. Here, $R[t_0,\dots ,t_n]$ is the polynomial ring in $n+1$ variables over R. When R is a field, these noncommutative Jordan algebras have highly degenerate norms, analogously as discussed in a similar construction for quadratic alternative algebras with degenerate norms, hence large radicals, employing Zorn algebras in ([7] Section 3.8).

We point out that the abstract colour algebras we study here originated from a problem in physics. In physics, the colour symmetry of the Gell–Mann quark model can be described as the multiplication of a colour algebra: For each quark, there is an antiquark which has the opposite properties of the quark. Quarks and antiquarks are used to form particles called hadrons. Each quark comes in three varieties, and colour was used to describe the interactions of quarks. For an accessible explanation of the construction of the resulting colour algebra, see [8]. Since split octonions also play a role in particle physics (for a beautiful overview, cf. [9]), and are directly related to Penrose’s classical twistors, we conjecture that (the symmetries of) colour algebras [2] over the real and complex numbers might play a new unexpected role in other areas of physics as well.

More precisely, we can describe the split colour algebra $\mathrm{Col}\left(F\right)$ as the seven-dimensional algebra with basis $1,u_i,v_i$ for $1\le i\le 3$, such that $u_i{u}_j={\epsilon }_{i,j,k}{v}_k$, $v_i{v}_j={\epsilon }_{i,j,k}{u}_k$, and $u_i{v}_j=v_i{u}_j={\delta }_{ij}1$, where ${\epsilon }_{i,j,k}$ is the totally skew-symmetric tensor with ${\epsilon }_{1,2,3}=1$. This algebra describes the colour symmetries of the quark model [10] when $F=\mathbb{C}$.

Under this point of view, a colour algebra A over F is then a form of the split colour algebra $\mathrm{Col}\left(F\right)$, that means there exists a field extension $K/F$ such that $A{\otimes }_{F}K\cong \mathrm{Col}\left(F\right)$.

2. Preliminaries

Let R be a unital commutative associative ring and let $R^{\times }$ denote the set of invertible elements in R. We will assume that $2\in R^{\times }$. For $P\in \mathrm{Spec}\left(R\right)$ let $R_P$ be the localisation of R at P and $m_P$ the maximal ideal of $R_P$. We denote the corresponding residue class field by $k\left(P\right)=R_P/m_P$. For an R-module M, the localisation of M at P is denoted by $M_P$. An R-module M has full support if $M_P\ne 0$ for all $P\in \mathrm{Spec}\left(R\right)$. All nonassociative R-algebras A considered in this paper are finitely generated projective of constant rank as an R-module and have full support.

A unital algebra A over R is called a colour algebra if $A\left(P\right)=A\otimes k\left(P\right)$ is a colour algebra over the field $k\left(P\right)$ for all $P\in \mathrm{Spec}\left(R\right)$.

Remark 1.

There are several standard ways to define a class of algebras that has previously only been defined over fields. If the class is characterised by a set of identities, one can use these identities. This is sometimes called a “global” definition. However, even for base fields, the author is not aware of any set of defining identities for colour algebras, or some other characterisation of their multiplicative structure that lends itself to be used to define these algebras also over rings. The identities that define the split colour algebra $\mathrm{Col}\left(F\right)$ are much too restrictive and would already miss out the more general split colour algebras that can be defined over the non-free modules that we describe in Section 3.1. To use the identities given in ([2] Theorem 3.1) to define a colour algebra as a quadratic algebra with a nondegenerate norm form whose crossed product(s) satisfy certain equations is also too restricitve, as it partly relies on the existence of linearly independent elements, a concept that makes little sense if we do not want to limit us to algebras that are only defined on free R-modules.

The “local” way to define a class of algebras over rings using fibres, i.e., the one we use here, is consistent with standard approaches from algebraic geometry. We believe that this approach provides a natural definition of a colour algebra over a ring. It seems also the most general one, as it makes no assumptions on the module structure of the algebra or its subalgebras.

Alternatively, one might also try to define a colour algebra as a form of a split colour algebra as defined in Section 3.1, i.e., an algebra that becomes isomorphic to a split colour algebra over some faithfully flat or étale extension of R; however, this approach would exclude (potentially existing) colour algebras which have absolutely indecomposable underlying R-module structures.

A unital algebra A is called quadratic, if there exists a quadratic form $n:A\to R$ such that $n\left(1_A\right)=1$ and $x^2-n(1_A,x)x+n\left(x\right)1_A=0$ for all $x\in A$, where $n(x,y)$ denotes the induced symmetric bilinear form $n(x,y)=n(x+y)-n\left(x\right)-n\left(y\right)$. The form n is uniquely determined and called the norm of the quadratic algebra A [7].

An anti-automorphism $\sigma :A\to A$ of period two is called an involution on A. An involution $\sigma$ is called scalar if $\sigma \left(x\right)x\in R1$ for all $x\in A$. For every scalar involution $\sigma$, the norm $n_A:A\to R$, $n_A\left(x\right)=\sigma \left(x\right)x$ (resp. the trace $t_A:A\to R$, $t_A\left(x\right)=\sigma \left(x\right)+x$) is a quadratic (resp. an R-linear) form. If an algebra A has a scalar involution then A is a quadratic algebra [11].

If A is a quadratic algebra over R, then $A=R1\oplus A_0$ with $A_0=\{u\in A|t\left(u\right)=0\}$, since we assume that $2\in R^{\times }$ is invertible.

Define $\times :A_0\times A_0\to A_0$, $u\times v=pr\left(uv\right)$ with $pr:A\to A_0$ the canonical projection map. Then $(A_0,\times )$ is an anticommutative algebra over R with $uv=-{\displaystyle \frac{1}{2}}n(u,v)+u\times v$. The algebra $(A_0,\times )$ is called the vector algebra of A.

A unital algebra C over R is called a composition algebra if there exists a quadratic form $n:C\to R$ which satisfies the following two conditions. (i) The quadratic form is multiplicative, which means it satisfies $n\left(xy\right)=n\left(x\right)n\left(y\right)$ for all $x,y\in C$. (ii) The symmetric bilinear form $n(x,y)=n(x+y)-n\left(x\right)-n\left(y\right)$ which is induced by n is nondegenerate, i.e., it determines an R-module isomorphism $C\stackrel{\sim }{⟶}\check{C}={\mathrm{Hom}}_R(C,R)$. The quadratic form n is called the norm of C and is uniquely determined up to isometry [7]. Every composition algebra over a ring has either rank 1, 2, 4, or 8. Composition algebras of rank 2 are exactly the quadratic étale algebras over R (these are sometimes called tori in the literature). The composition algebras of rank four are called quaternion algebras, and they are Azumaya algebras of rank four. Composition algebras of rank eight are called octonion algebras. A composition algebra C over R is called split if C contains an isomorphic copy of the split torus $R\times R$ with isotropic norm $n\left(\right(a,b\left)\right)=ab$ as a composition subalgebra.

Every composition algebra C has a canonical involution $\overline{\phantom{C}}$ given by $\overline{x}=t\left(x\right)1_C-x$, where $t:C\to R$, $t\left(x\right)=n(1_C,x)$ is the trace of C. This involution is scalar. We know that $n\left(x\right)=x\overline{x}$ for all $x\in C$.

In our setup $2\in R^{\times }$, all the octonion algebras over R with a quadratic étale subalgebra can be constructed employing a hermitian space of rank three with trivial determinant [12].

Let $\times :R^3\times R^3\to R^3$ denote the classical vector product on the three-dimensional column space $R^3$. The algebra

$$\mathrm{Zor}\left(R\right)=\left[\begin{array}{cc}R& R^3\\ R^3& R\end{array}\right],$$

$$\left[\begin{array}{cc}a& u\\ u^{\prime }& a^{\prime }\end{array}\right]\left[\begin{array}{cc}b& v\\ v^{\prime }& b^{\prime }\end{array}\right]=\left[\begin{array}{cc}ab{+}^{t}u{v}^{\prime }& av+b^{\prime }u-u^{\prime }\times v^{\prime }\\ b{u}^{\prime }+a^{\prime }{v}^{\prime }+u\times v& {}^{t}u^{\prime }v+a^{\prime }{b}^{\prime }\end{array}\right],$$

is a split octonion algebra over R with norm and trace given by

$$\det \left[\begin{array}{cc}a& u\\ u^{\prime }& a^{\prime }\end{array}\right]=a{a}^{\prime }{-}^{t}u{u}^{\prime },\mathrm{tr}\left(\left[\begin{array}{cc}a& u\\ u^{\prime }& a^{\prime }\end{array}\right]\right)=a+a^{\prime },$$

and is called Zorn’s algebra of vector matrices. If R is a field or more generally, a principal ideal domain or a Dedekind domain, then $\mathrm{Zor}\left(R\right)$ is, up to the isomorphism, the only split octonion algebra over R.

We include Table 1 to help the reader with the terminology used throughout.

Table 1. Notation.

Symbol	Definition
R	A unital commutative ring with $2\in R^{\times }$.
$\check{P}$	The dual module ${Hom}_R(P,R)$.
$\mathrm{Col}\left(R\right)$	The classical split colour algebra over the base ring R.
$\mathrm{Zor}\left(R\right)$	Zorn’s classical split octonion algebra of vector matrices over R.
$(T,\alpha )$	A proj. R-module T of constant rank 3, an isomorphism $\alpha :{\bigwedge }^3\left(T\right)\to R$.
$\mathrm{Col}(T,\alpha )$	The split colour algebra constructed from $(T,\alpha )$.
$\mathrm{Zor}(T,\alpha )$	The Zorn algebra constructed from $(T,\alpha )$ (a split octonion algebra).
$W(T,\alpha )$	The split vector colour algebra constructed from $(T,\alpha )$.
$\mathrm{Col}(S,P,h,\alpha )$	The colour algebra constructed from a hermitian form $h:P\times P\to S$.
$\mathrm{Cal}(S,P,h,\alpha )$	The octonion algebra constructed from a hermitian form $h:P\times P\to S$.
$W(S,P,h,\alpha )$	The vector algebra constructed from a hermitian form $h:P\times P\to S$.

3. Constructions of Colour Algebras over Rings

3.1. Split Colour Algebras over Rings

Let T be a projective R-module of constant rank 3 such that $\det T={\bigwedge }^3\left(T\right)\cong R$. Let $⟨,⟩:T\times \check{T}\to R,$ $⟨u,\check{v}⟩=\check{v}\left(u\right)$ be the canonical pairing between T and its dual module $\check{T}={\mathrm{Hom}}_R(T,R)$. Every isomorphism $\alpha :{\bigwedge }^3\left(T\right)\to R$ induces a bilinear map $\times :T\times T\to \check{T}$ via

$$(u,v)\mapsto u\times v=\alpha (u\wedge v\wedge -).$$

Such a map is called a vector product on T since locally it is the classical vector product. Moreover, $\alpha$ also determines an isomorphism $\beta :\det \check{T}\to R$ which satisfies

$$\alpha (u_1\wedge u_2\wedge u_3)\beta ({\check{u}}_1\wedge {\check{u}}_2\wedge {\check{u}}_3)=\det \left(⟨u_i,{\check{u}}_j⟩\right)$$

for all $u_i\in T$, ${\check{u}}_j\in \check{T}$, $1\le i,j\le 3$. Therefore, we analogously obtain a vector product $\check{T}\times \check{T}⟶T$ on $\check{T}$ via

$$(\check{u},\check{v})\mapsto \check{u}\times \check{v}=\beta (\check{u}\wedge \check{v}\wedge -),$$

employing $\beta$ instead of $\alpha$.

For these generalised vector products we have $u\times u=0$ for all $u\in T$, and $\check{u}\times \check{u}=0$ for all $\check{u}\in \check{T}$ since this holds locally for the classical vector products.

Consider the finitely generated projective R-module of constant rank seven

$$\mathrm{Col}(T,\alpha )=R\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\oplus T\oplus \check{T}=\{\left[\begin{array}{cc}a& 0\\ 0& a\end{array}\right]|a\in R\}\oplus \{\left[\begin{array}{cc}0& u\\ \check{u}& 0\end{array}\right]|u\in T,\check{u}\in \check{T}\}$$

$$=\{\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right]|a\in R,u\in T,\check{u}\in \check{T}\}$$

together with the multiplication given by

$$\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right]\left[\begin{array}{cc}b& v\\ \check{v}& b\end{array}\right]=\left[\begin{array}{cc}ab+{\displaystyle \frac{1}{2}}(⟨u,\check{v}⟩+⟨v,\check{u}⟩)& av+bu-\check{u}\times \check{v}\\ b\check{u}+a\check{v}+u\times v& ab+{\displaystyle \frac{1}{2}}(⟨u,\check{v}⟩+⟨v,\check{u}⟩)\end{array}\right].$$

The algebra $\mathrm{Col}(T,\alpha )$ is a colour algebra and is called a split colour algebra over R. A split colour algebra $\mathrm{Col}(T,\alpha )$ over R has the unit element $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, and norm and trace given by

$$n(\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right])=a^2-⟨u,\check{u}⟩,\mathrm{tr}(\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right])=2a.$$

Lemma 1.

The split colour algebra $\mathrm{Col}(T,\alpha )$ is a quadratic and flexible algebra over R, and hence a noncommutative Jordan algebra.

In particular, $\mathrm{Col}(T,\alpha )$ is power associative.

Proof. 

Let $x\in \mathrm{Col}(T,\alpha )$, that is $x=\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right]$.

Using the identities $\check{u}\times \check{u}=0$ and $u\times u=0$, we obtain that

$$x^2=\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right]\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right]$$

$$=\left[\begin{array}{cc}{a}^2+{\displaystyle \frac{1}{2}}(⟨u,\check{u}⟩+⟨u,\check{u}⟩)& 2au-\check{u}\times \check{u}\\ a\check{u}+a\check{u}+u\times u& a^2+{\displaystyle \frac{1}{2}}(⟨u,\check{u}⟩+⟨u,\check{u}⟩)\end{array}\right]$$

$$=\left[\begin{array}{cc}{a}^2+⟨u,\check{u}⟩& 2au\\ a\check{u}+a\check{u}& a^2+⟨u,\check{u}⟩\end{array}\right]$$

and

$$\mathrm{tr}\left(x\right)x=\left[\begin{array}{cc}2a^2& 2au\\ 2a\check{u}& 2a^2\end{array}\right],$$

and thus

$$x^2-\mathrm{tr}\left(x\right)x+n\left(x\right)1$$

$$=\left[\begin{array}{cc}{a}^2+⟨u,\check{u}⟩& 2au-\check{u}\times \check{u}\\ a\check{u}+a\check{u}+u\times u& a^2+⟨u,\check{u}⟩\end{array}\right]-\left[\begin{array}{cc}2a^2& 2au\\ 2a\check{u}& 2a^2\end{array}\right]+\left[\begin{array}{cc}{a}^2-⟨u,\check{u}⟩& 0\\ 0& a^2-⟨u,\check{u}⟩\end{array}\right]$$

$$=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right].$$

To show flexibility, that is $[x,y,x]=0$ for all $x,y\in \mathrm{Col}(T,\alpha )$, we observe that the algebra is of the form $(R,T\oplus \widehat{T},B,\times )$, with $B(u,v)={\displaystyle \frac{1}{2}}(h(u,v)+\overline{h(u,v)})$ as defined in ref. [7], so that flexibility follows from ([7] Example 2, Lemma 2) (or alternatively, from the identities in ([12] Theorem 2.1)). Note that it suffices to show flexibility for all elements with diagonal elements zero.

Since $x^2=tr\left(x\right)x-n\left(x\right)1$ and the associator is trilinear, we get $[x,y,x^2]=[x,y,tr\left(x\right)x-n\left(x\right)1]=tr\left(x\right)[x,y,x]-n\left(x\right)[x,y,1]=tr\left(x\right)[x,y,x].$ Since $[x,y,x]=0$ for all $x,y\in \mathrm{Col}(R\times R,T,\alpha )$, the Jordan identity $[x,y,x^2]=0$ thus holds for all $x,y\in \mathrm{Col}(T,\alpha )$ ([11] (3.3)). □

Projecting the multiplication of $\mathrm{Col}(T,\alpha )$ onto the submodule $T\oplus \check{T}$ yields the six-dimensional split vector colour algebra $W(T,\alpha )=W(R\times R,T\oplus \check{T},h,\alpha )=T\oplus \check{T}$ over R with multiplication

$$(u,\check{u})(v,\check{v})=(-\check{u}\times \check{v},u\times v).$$

The multiplication of $W(T,\alpha )$ is anticommutative and satisfies the identity

$$\left(\left((u,\check{u})(v,\check{v})\right)(v,\check{v})\right)(v,\check{v}))=n\left((v,\check{v})\right)(u,\check{u})(v,\check{v})$$

with $n\left((v,\check{v})\right)=-⟨v,\check{v}⟩,$ analogously as in ref. [1] (and adjusted by the factor ${\displaystyle \frac{1}{2}}$ because of our different definition of the norm).

Split colour algebras are closely connected to split octonion algebras, as the split vector colour algebra $W(T,\alpha )$ of the split colour algebra $\mathrm{Col}(T,\alpha )$ is the same as the algebra we obtain when projecting the multiplication of the Zorn algebra $\mathrm{Zor}(T,\alpha )$ to $T\oplus \check{T}$. Here,

$$\mathrm{Zor}(T,\alpha )=\left[\begin{array}{cc}R& T\\ \check{T}& R\end{array}\right]$$

with the multiplication

$$\left[\begin{array}{cc}a& u\\ \check{u}& a^{\prime }\end{array}\right]\left[\begin{array}{cc}b& v\\ \check{v}& b^{\prime }\end{array}\right]=\left[\begin{array}{cc}ab+⟨u,\check{v}⟩& av+b^{\prime }u-\check{u}\times \check{v}\\ b\check{u}+a^{\prime }\check{v}+u\times v& ⟨v,\check{u}⟩+a^{\prime }{b}^{\prime }\end{array}\right]$$

is a split octonion algebra over R with norm

$$\det \left[\begin{array}{cc}a& u\\ \check{u}& a^{\prime }\end{array}\right]=a{a}^{\prime }-⟨u,\check{u}⟩.$$

The octonion algebra $\mathrm{Zor}(T,\alpha )$ is called a Zorn algebra. Every split octonion algebra over R is isomorphic to such a Zorn algebra. Locally, $\mathrm{Zor}(T,\alpha )$ looks like $\mathrm{Zor}\left(R\right)$ ([7] Sections 3.3 and 3.4, Theorem 3.5).

It is now straightforward to see that every split colour algebra $\mathrm{Col}(T,\alpha )$ can be constructed from the vector part $(T,\alpha )$ of a Zorn algebra $\mathrm{Zor}(T,\alpha )$ over R.

When $T=R^3$ and ${\times }_{\alpha }=\times :R^3\times R^3\to R^3$ is the classical vector product on $R^3$, then the split colour algebra over R can be written as

$$\mathrm{Col}\left(R\right)=\left[\begin{array}{cc}R& R^3\\ R^3& R\end{array}\right],$$

$$\left[\begin{array}{cc}a& u\\ u^{\prime }& a\end{array}\right]\left[\begin{array}{cc}b& v\\ v^{\prime }& b\end{array}\right]=\left[\begin{array}{cc}ab+{\displaystyle \frac{1}{2}}{(}^{t}u{v}^{\prime }{+}^t{u}^{\prime }v)& av+bu-u^{\prime }\times v^{\prime })\\ b{u}^{\prime }+a{v}^{\prime }+u\times v& ab+{\displaystyle \frac{1}{2}}{(}^{t}u{v}^{\prime }{+}^t{u}^{\prime }v)\end{array}\right],$$

with norm

$$\det \left[\begin{array}{cc}a& u\\ u^{\prime }& a\end{array}\right]=a^2{-}^{t}u{u}^{\prime }.$$

Again, the split colour algebra $\mathrm{Col}\left(R\right)$ can be constructed from the vector part of $\mathrm{Zor}\left(R\right)$. Locally, $\mathrm{Col}(T,\alpha )$ looks like $\mathrm{Col}\left(R\right)$.

The constructions of $\mathrm{Col}(T,\alpha )$, $\mathrm{Zor}(T,\alpha )$ and $W(T,\alpha )$ are functorial in the parameters involved. Suppose that $T,T^{\prime }$ are two projective R-modules of constant rank 3 such that $\det T\cong \det T^{\prime }\cong R$, and such that $\alpha :{\bigwedge }^3\left(T\right)\to R$, ${\alpha }^{\prime }:{\bigwedge }^3\left(T^{\prime }\right)\to R$ are two isomorphisms. An R-linear map $\phi :T\to T^{\prime }$ such that ${\alpha }^{\prime }\circ det\left(\phi \right)=\alpha$ is called a morphism between $(T,\alpha )$ and $(T^{\prime },{\alpha }^{\prime })$ and denoted by $(T,\alpha )\to (T^{\prime },{\alpha }^{\prime })$. If $\phi :(T,\alpha )\to (T^{\prime },{\alpha }^{\prime })$ is such a morphism then $\phi$ is bijective, and induces “diagonal” isomorphisms between the corresponding split colour and Zorn algebras:

Proposition 1.

If $\phi :(T,\alpha )\to (T^{\prime },{\alpha }^{\prime })$ is a morphism, then

$$\mathrm{Col}(T,\alpha )\cong \mathrm{Col}(T^{\prime },{\alpha }^{\prime })\mathit{via}$$

$$\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right]\mapsto \left[\begin{array}{cc}a& \phi \left(u\right)\\ {\check{\phi }}^{-1}\left(\check{u}\right)& a\end{array}\right],$$

$$\mathrm{Zor}(T,\alpha )\cong \mathrm{Zor}(T^{\prime },{\alpha }^{\prime })\mathit{via}$$

$$\left[\begin{array}{cc}a& u\\ \check{u}& b\end{array}\right]\mapsto \left[\begin{array}{cc}a& \phi \left(u\right)\\ {\check{\phi }}^{-1}\left(\check{u}\right)& b\end{array}\right],$$

and

$$\mathrm{Col}(T,\alpha )\cong \mathrm{Col}(\check{T},\beta )\mathit{via}$$

$$\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right]\mapsto {\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right]}^{*t}=\left[\begin{array}{cc}a& -\check{u}\\ -u& a\end{array}\right]$$

$$\mathrm{Zor}(T,\alpha )\cong \mathrm{Zor}(\check{T},\beta )\mathit{via}$$

$$\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right]\mapsto {\left[\begin{array}{cc}a& u\\ \check{u}& b\end{array}\right]}^{*t}=\left[\begin{array}{cc}b& -\check{u}\\ -u& a\end{array}\right]$$

This is proved for Zorn algebras in ([7] Section 3.4). The proof is analogous for the split colour algebras.

Thus $(T,\alpha )$ determines the algebras $\mathrm{Zor}(T,\alpha )$ and $\mathrm{Col}(T,\alpha )$ up to isomorphism, as well as the associated six-dimensional split vector colour algebra $W(T,\alpha )$.

In particular, if R contains a primitive third root of unity $\nu$, then there exist two nontrivial automorphisms in $\mathrm{Zor}(T,\alpha )$ and $\mathrm{Col}(T,\alpha )$ induced by the two primitive third roots of unity in R (cf. [7] for the proof for $\mathrm{Zor}(T,\alpha )$).

Corollary 1.

Suppose that there exists an R-linear map $\phi :T\to T$ such that $\det \left(\phi \right)=id$. Then

$$\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right]\mapsto \left[\begin{array}{cc}a& \phi \left(u\right)\\ {\check{\phi }}^{-1}\left(\check{u}\right)& a\end{array}\right]$$

lies in ${\mathrm{Aut}}_R\left(\mathrm{Col}(T,\alpha )\right)$. Let $\nu \in R^{\times }$ such that ${\nu }^3=1$. Then

$$\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right]\to \left[\begin{array}{cc}a& \nu u\\ {\mu }^{-1}\left(\check{u}\right)& a\end{array}\right].$$

lies in ${\mathrm{Aut}}_R\left(\mathrm{Col}(T,\alpha )\right)$.

3.2. A Construction of Colour Algebras Employing Hermitian Forms

The construction of split colour algebras is part of a bigger picture. When $2\in R^{\times }$, every octonion algebra which contains a quadratic étale subalgebra can be constructed employing a nondegenerate hermitian form of rank three with trivial determinant (Petersson and Racine ([13] Section 3.8), or Thakur [12]). We now link this construction to colour algebras.

Let S be a quadratic étale R-algebra with canonical involution $\overline{\phantom{C}}$ and let P be a finitely generated projective R-module of constant rank. A $\overline{\phantom{C}}$-hermitian form $h:P\times P\to S$ is a biadditive map with $h(ws,w^{\prime }t)=\overline{s}h(w,w^{\prime })t$ and $h(w,w^{\prime })=\overline{h(w^{\prime },w)}$ for all $s,t\in S$, $w,w^{\prime }\in P$, and where $P\to {\overline{P}}^{\vee }$, $w\mapsto h(w,·)$ is an isomorphism of S-modules.

Let P have rank three and let $(P,h)$ be a nondegenerate $\overline{\phantom{C}}$-hermitian space such that ${\bigwedge }^3(P,h)\cong ⟨1⟩$, where $⟨1⟩$ is the $\overline{\phantom{C}}$-hermitian form $h_0(a,b)=a\overline{b}$ on S. Then

$$n:P\times P\to R,n(u,v)=h(u,v)+\overline{h(u,v)}$$

is a nondegenerate symmetric R-bilinear form.

Choose an S-isomorphism $\alpha :{\bigwedge }^3(P,h)\to ⟨1⟩$. Define a cross product ${\times }_{\alpha }:P\times P\to P$ via

$$h(u,v{\times }_{\alpha }w)=\alpha (u\wedge v\wedge w)$$

for all $u,v,w\in P$ as in ([12] p. 5122) or ([13] Section 3.8). Note that this cross product ${\times }_{\alpha }$ depends only on $T,P,h$ and $\alpha$. The R-module $C=\mathrm{Col}(S,P,h,\alpha )=R\oplus P$ becomes a colour algebra under the multiplication

$$(a,u)(b,v)=(ab-{\displaystyle \frac{1}{2}}(h(u,v)+\overline{h(u,v)}),va+ub+u{\times }_{\alpha }v)$$

for all $u,v\in P$ and $a,b\in R$. The colour algebra $\mathrm{Col}(S,P,h,\alpha )$ is a flexible and quadratic algebra with norm

$$n\left((a,u)\right)=a^2+h(u,u),$$

and

$$n((a,u),(b,v))=n_S(a,b)+(h(u,v)+\overline{h(u,v)})=(a,u)\sigma (b,v)+(b,v)\sigma (a,u)$$

with $\sigma (a,u)=(\overline{a},-u)$.

In $\mathrm{Col}(S,P,h,\alpha )$ we have

$$\left(\right(u\times v)\times v)\times v=n\left(v\right)u\times v$$

for all $u,v\in P$ ([14] Lemma 2). It is straightforward to check that

$$a\left(u{\times }_{\alpha }v\right)=\left(\overline{a}u\right){\times }_{\alpha }v=u{\times }_{\alpha }\overline{a}v,$$

$$h(u,v{\times }_{\alpha }w)=\overline{h(u{\times }_{\alpha }v,w)}=h(w,u{\times }_{\alpha }v),$$

$$h(u{\times }_{\alpha }v,u)=h(u{\times }_{\alpha }v,v)=0$$

for all $u,v,w\in P$, $a\in S$.

We obtain split colour algebras as the special case that $S=R\times R$ is split and $P\cong T\oplus \check{T}$ as R-module. In that case a hermitian form h on P is induced by the R-bilinear form

$$b:(T\oplus \check{T})\times (T\oplus \check{T})\to R,b((u,\check{u}),(v,\check{v}))=⟨u,\check{v}⟩+⟨v,\check{u}⟩.$$

If we project the multiplication of $\mathrm{Col}(S,P,h,\alpha )$ to P, we obtain the map $pr:\mathrm{Col}(S,P,h,\alpha )\times \mathrm{Col}(S,P,h,\alpha )\to P$,

$$(a,u)(b,v)=(a,u)(b,v)=va+ub+u{\times }_{\alpha }v$$

for all $u,v\in P$ and $a,b\in S$, and then restricting this map to $P\times P$ we obtain the R-bilinear map $P\times P\to P$, $(u,v)=u{\times }_{\alpha }v$. This makes P into a noncommutative non-unital six-dimensional algebra $(P,{\times }_{\alpha })$ over R which we also denote by $W(S,P,h,\alpha )$ and call a vector colour algebra. Since its multiplication is given by the cross product $u{\times }_{\alpha }v,$ locally it is a classical vector colour algebra with the classical cross product.

Remark 2.

(i) We recall that the R-module $\mathrm{Cay}(S,P,h,\alpha )=S\oplus P$ becomes an octonion algebra under the multiplication

$$(a,u)(b,v)=(ab-h(v,u),va+u\overline{b}+u{\times }_{\alpha }v)$$

for $u,v\in P$ and $a,b\in S$, with norm

$$n\left((a,u)\right)=n_S\left(a\right)+h(u,u).$$

Its canonical involution $\sigma (a,u)=(\overline{a},-u)$ is a scalar involution. Its norm can also be written as $n\left(\right(a,u\left)\right)=(a,u)\sigma (a,u)=\sigma (a,u)(a,u)$ and its trace as $t\left((a,u)\right)=\sigma (a,u)+(a,u)=(t_S\left(a\right),0)$.

(ii) We have the following identities in the octonion algebra $A=\mathrm{Cay}(S,P,h,\alpha )$ [12,14]:

$$h(u{\times }_{\alpha }v,u)+\overline{h(u{\times }_{\alpha }v,u)}=n_A(u{\times }_{\alpha }v,u)=0,$$

$$\left(u{\times }_{\alpha }v\right){\times }_{\alpha }u=u\times \left(v{\times }_{\alpha }u\right),$$

$$h(u,u{\times }_{\alpha }v)=0\mathit{and}u{\times }_{\alpha }\left(u{\times }_{\alpha }v\right)=-h(u,u)v+h(v,u)u,$$

$$h(u{\times }_{\alpha }v,v)=0\mathit{and}\left(u{\times }_{\alpha }v\right){\times }_{\alpha }v=h(v,v)u-h(u,v)v$$

for all $u,v\in P$. Observe that projecting the multiplication of $A=\mathrm{Cay}(S,P,h,\alpha )$ onto P, we obtain the map $pr:A\times A\to P$,

$$(a,u)(b,v)=(0,va+u\overline{b}+u{\times }_{\alpha }v)$$

for all $u,v\in P$ and $a,b\in S$, and then restricting this map to $P\times P$ we obtain the R-bilinear map $P\times P\to P$, $(u,v)=u{\times }_{\alpha }v$, i.e., also the algebra $W(S,P,h,\alpha )=(P,{\times }_{\alpha })$.

(iii) There exist octonion algebras whose norm restricted to their trace zero elements forms an indecomposable quadratic space of rank 7 [15], and thus have no proper composition subalgebra. A prominent example is Coxeter’s order over $\mathbb{Z}$ which does not have any proper composition subalgebra. This is contrary to the situation of colour algebras over fields, and hence not every octonion algebra over a ring can be used to construct a colour algebra as described above.

(iv) At the other extreme, when every octonion algebra over R contains a quadratic étale algebra S then every octonion algebra yields a colour algebra $\mathrm{Col}(S,P,h,\alpha )$.

This is the case for example, when R is a semilocal ring with $2\in R^{\times }$ [16].

4. Isomorphisms, Automorphisms and Derivations

Let $S_1$ and $S_2$ be two quadratic étale algebras over R with nontrivial automorphism ${\overline{\phantom{C}}}_1$ and ${\overline{\phantom{C}}}_2$, and let $(P,h)$ and $(P^{\prime },h^{\prime })$ be two nondegenerate ${\overline{\phantom{C}}}_i$-hermitian spaces over $S_i$, $i\in \{1,2\}$. Then $(P,h)$ and $(P^{\prime },h^{\prime })$ are called isometric, if there exists an R-isomorphism $s:S_1\to S_2$ and an s-semilinear map $\mathsf{\Psi }:P\to P^{\prime }$ such that $h^{\prime }(\mathsf{\Psi }\left(u\right),\mathsf{\Psi }\left(v\right))=s\left(h(u,v)\right)$ for all $u,v\in P$. We now generalize ([3] Theorem 2.5):

Proposition 2.

Let $(P,h)$ and $(P^{\prime },h^{\prime })$ be two nondegenerate ${\overline{\phantom{C}}}_i$-hermitian spaces over $S_i$ of rank three with trivial determinant $i\in \{1,2\}$. Let $n\left(v\right)=h(v,v)$ and $n^{\prime }\left(v\right)=h^{\prime }(v,v)$ be the associated symmetric bilinear forms on P and $P^{\prime }$.

(i) 

If $(P,h)$ and $(P^{\prime },h^{\prime })$ are isometric then $(P,{\times }_{\alpha })=W(S_1,P,h,\alpha )$ and $(P^{\prime },{\times }_{\alpha }^{\prime })=W(S_2,P^{\prime },h^{\prime },{\alpha }^{\prime })$ are isomorphic vector colour algebras.

(ii) 

If $f:W(S_1,P,h,\alpha )\to W(S_2,P^{\prime },h^{\prime },{\alpha }^{\prime })$ is an R-algebra isomorphism then n and $n^{\prime }$ are isometric quadratic forms over R.

Proof. 

(i) If $(P,h)$ and $(P^{\prime },h^{\prime })$ are isometric, then by definition there exists an R-algebra isomorphism $s:S_1\to S_2$ and an s-semilinear map $f:P\to P^{\prime }$ that satisfies $h^{\prime }(f\left(u\right),f\left(v\right))=s\left(h(u,v)\right)$ and ${\alpha }^{\prime }(f\left(u\right),f\left(v\right),f\left(w\right))=\alpha (u,v,w)$ for all $u,v,w\in P$. By assumption, we obtain that

$$h^{\prime }(f\left(u\right),f\left(v\right){\times }_{{\alpha }^{\prime }}f\left(w\right))={\alpha }^{\prime }(f\left(u\right)\wedge f\left(v\right)\wedge f\left(w\right)))$$

$$=s\left(\alpha (u,v,w)\right)=h(u,v{\times }_{\alpha }w)=h^{\prime }(f\left(u\right),f\left(v\right){\times }_{\alpha }f\left(w\right))$$

for all $u,v,w\in P$. Since $h^{\prime }$ is nondegenerate, this implies that f is an isomorphism between the vector algebras $(P,{\times }_{\alpha })=W(S_1,P,h,{\times }_{\alpha })$ and $(P^{\prime },{\alpha }^{\prime })=W(S_2,P^{\prime },h^{\prime },{\alpha }^{\prime })$.

(ii) Suppose that $f:W(S_1,P,h,\alpha )\to W(S_2,P^{\prime },h^{\prime },{\alpha }^{\prime })$ is an algebra isomorphism. We know that we have

$$\left(\right(u\times v)\times v)\times v=n\left(v\right)u\times v$$

in $\mathrm{Col}(S_1,P,h,\alpha )$ for all $u,v\in P$ ([14] Lemma 2) hence also in $W(S_1,P,h,\alpha )$. Apply f to this relation to obtain that $h^{\prime }(f\left(v\right),f\left(v\right))=h(v,v)$ for all $v\in P$, i.e., n and $n^{\prime }$ are isometric quadratic forms. □

Suppose now that $(P,h)$ and $(P^{\prime },h^{\prime })$ are two nondegenerate $\overline{\phantom{C}}$-hermitian spaces of rank three over a quadratic étale algebra S with trivial determinant. Choose two isomorphisms $\alpha :{\bigwedge }^3(P,h)\to ⟨1⟩$, ${\alpha }^{\prime }:{\bigwedge }^3(P^{\prime },h^{\prime })\to ⟨1⟩$. Define two cross products ${\times }_{\alpha }:P\times P\to P$,

$$h(u,v\times w)=\alpha (u\wedge v\wedge w)$$

and ${\times }_{{\alpha }^{\prime }}:P^{\prime }\times P^{\prime }\to P^{\prime }$,

$$h^{\prime }(u^{\prime },v^{\prime }\times w^{\prime })={\alpha }^{\prime }(u^{\prime }\wedge v^{\prime }\wedge w^{\prime }).$$

Let $\phi :P\to P^{\prime }$ be an S-linear map such that ${\alpha }^{\prime }\circ det\left(\phi \right)=\alpha$, i.e., $\phi$ is a (bijective) morphism $(P,\alpha )\to (P^{\prime },{\alpha }^{\prime })$. Assume additionally that $\phi :(P,\alpha )\to (P^{\prime },{\alpha }^{\prime })$ is an isometry between h and $h^{\prime }$.

Proposition 3.

Let $(P,h)$ and $(P^{\prime },h^{\prime })$ be two nondegenerate $\overline{\phantom{C}}$-hermitian spaces of rank three over a quadratic étale algebra S with a trivial determinant. Let $\alpha :{\bigwedge }^3(P,h)\to ⟨1⟩$ and ${\alpha }^{\prime }:{\bigwedge }^3(P^{\prime },h^{\prime })\to ⟨1⟩$ be two isomorphisms. Suppose that there is a morphism $\phi :(P,\alpha )\to (P^{\prime },{\alpha }^{\prime })$ that is an isometry between h and $h^{\prime }$. Then the following holds:

(i) 

$\phi \left(u{\times }_{\alpha }v\right)=u{\times }_{{\alpha }^{\prime }}v$.

(ii) 

$W(S,P,h,\alpha )\cong W(S,P^{\prime },h^{\prime },{\alpha }^{\prime })$ via φ.

(iii) 

$\mathrm{Col}(S,P,h,\alpha )\cong \mathrm{Col}(S,P^{\prime },h^{\prime },{\alpha }^{\prime })$ via $G_{\phi }\left((a,u)\right)=(a,\phi \left(u\right))$.

(iv) 

$\mathrm{Cay}(S,P,h,\alpha )\cong \mathrm{Cay}(S,P^{\prime },h^{\prime },{\alpha }^{\prime })$ via $H_{\phi }\left((a,u)\right)=(a,\phi \left(u\right))$.

Proof. 

(i)  We have

$$h^{\prime }(\phi \left(u\right),\phi \left(v{\times }_{\alpha }w\right))=h(u,v{\times }_{\alpha }w)$$

$$=\alpha (u\wedge v\wedge w)={\alpha }^{\prime }(\phi \left(u\right)\wedge \phi \left(v\right)\wedge \phi \left(w\right))=h^{\prime }(\phi \left(u\right),\phi \left(v\right){\times }_{{\alpha }^{\prime }}\phi \left(w\right)).$$

Since h is nondegenerate, it follows that $\phi \left(u{\times }_{\alpha }v\right)=u{\times }_{{\alpha }^{\prime }}v$.

(ii)

It is trivial.

(iii)

It is now easy to see that for $G=G_{\phi }$, we have

$$G\left((a,u)(b,v)\right)=G(ab-{\displaystyle \frac{1}{2}}(h(u,v)+\overline{h(u,v)}),va+ub+u{\times }_{\alpha }v)$$

$$=(ab-{\displaystyle \frac{1}{2}}(h(u,v)+\overline{h(u,v)}),\phi \left(v\right)\phi \left(a\right)+\phi \left(u\right)\phi \left(b\right)+\phi \left(u{\times }_{\alpha }v\right))$$

and

$$G\left(\right(a,u\left)\right)G\left(\right(b,v\left)\right)=(a,\phi (u\left)\right)(b,\phi (v\left)\right)$$

$$=(ab-{\displaystyle \frac{1}{2}}(h^{\prime }(\phi \left(u\right),\phi \left(v\right))+\overline{h^{\prime }(\phi \left(u\right),\phi \left(v\right))}),\phi \left(v\right)a+\phi \left(u\right)b+u{\times }_{{\alpha }^{\prime }}v)$$

$$=(ab-{\displaystyle \frac{1}{2}}(h(u,v)+\overline{h(u,v)}),\phi \left(v\right))),\phi \left(v\right)a+\phi \left(u\right)b+u{\times }_{{\alpha }^{\prime }}v).$$

The second entries are equal due to our assumption that $\phi$ is S-linear.

(iv)

Analogously, we obtain for $H=H_{\phi }$:

$$H\left((a,u)(b,v)\right)=H(ab-h(u,v),va+u\overline{b}+u{\times }_{\alpha }v)$$

$$=(ab-h(u,v),\phi \left(v\right)\phi \left(a\right)+\phi \left(u\right)\phi \left(\overline{b}\right)+\phi \left(u{\times }_{\alpha }v\right))$$

and

$$H\left(\right(a,u\left)\right)H\left(\right(b,v\left)\right)=(a,\phi (u\left)\right)(b,\phi (v\left)\right)$$

$$=(ab-h^{\prime }(\phi \left(u\right),\phi \left(v\right)),\phi \left(v\right)a+\phi \left(u\right)\overline{b}+u{\times }_{{\alpha }^{\prime }}v)$$

$$=(ab-h(u,v),\phi \left(v\right))),\phi \left(v\right)a+\phi \left(u\right)\overline{b}+u{\times }_{{\alpha }^{\prime }}v).$$

  □

Corollary 2.

Let $(P,h)$ and $(P^{\prime },h^{\prime })$ be two nondegenerate $\overline{\phantom{C}}$-hermitian spaces of rank three over a quadratic étale algebra S with trivial determinant. Let $\alpha :{\bigwedge }^3(P,h)\to ⟨1⟩$ and ${\alpha }^{\prime }:{\bigwedge }^3(P^{\prime },h^{\prime })\to ⟨1⟩$ be two isomorphisms. If $\phi :(P,\alpha )\to (P^{\prime },{\alpha }^{\prime })$ is a morphism that is an isometry between h and $h^{\prime }$ such that $\det \left(\phi \right)=id$, then $G_{\phi }\in \mathrm{Aut}\left({\mathrm{Col}}_R(S,P,h,\alpha )\right)$ and $H_{\phi }\in {\mathrm{Aut}}_R\left(\mathrm{Cay}(S,P,h,\alpha )\right)$.

Remark 3.

If the ternary hermitian space $(P,h)$ is orthogonally decomposable, then the construction $\mathrm{Col}(S,P,h,\alpha )$ is independent of the choice of α. This is due to the fact that in that case, the corresponding octonion algebra $\mathrm{Cay}(S,P,h,\alpha )$ can be constructed by a generalised Cayley–Dickson doubling process out of a suitably chosen quaternion algebra, e.g., cf. ([14] Section 3), and hence so is $\mathrm{Col}(S,P,h,\alpha )$. We sketch how to proceed.

For composition algebras over rings, there exists a generalised Cayley–Dickson doubling process $\mathrm{Cay}(D,F,N_F)$ with which we can construct every composition algebra C that contains a composition algebra D of half its rank; see [7] for details. The underlying module structure for the algebra $\mathrm{Cay}(D,F,N_F)$ is given by $D\oplus F$ and its quadratic norm form is $N_C=N_D\perp (-N_F)$.

So when we start with an octonion algebra $C=\mathrm{Cay}(S,P_1\oplus P_2,h_1\perp h_2,\alpha )$ where the rank three hermitian form $(P,h)\cong (P_1,h_1)\oplus (P_2,h_2)$ decomposes as a hermitian form into a rank one nondegenerate hermitian form $(P_1,h_1)$ and a rank two nondegenerate hermitian form $(P_2,h_2)$, we can define a quaternion subalgebra $D=\mathrm{Cay}(S,P_1,-N_1)$ of C with $N_1\left(u\right)={\displaystyle \frac{1}{2}}{h}_1(u,u)$ and can write C uniquely as generalised Cayley–Dickson doubling

$$\mathrm{Cay}(S,P_1\oplus P_2,h_1\perp h_2,\alpha )=\mathrm{Cay}(D,P_1,-N_{P_1})$$

where $N_{P_1}={\displaystyle \frac{1}{2}}{h}_2(u,u).$ As a direct consequence, the part of the multiplication of C corresponding to $u{\times }_{\alpha }v$ is uniquely determined, hence independent of the chosen α. Hence, so is the multiplication of $\mathrm{Col}(S,P,h,\alpha )$.

Theorem 2.

Let S be a quadratic étale algebra over R. Then the maps $W(S,P,h,\alpha )\mapsto \mathrm{Cay}(S,P,h,\alpha )$ and $W(S,P,h,\alpha )\mapsto \mathrm{Col}(S,P,h,\alpha )$ induce bijections between the following sets:

(i) 

The set of all pairs $\left(\mathrm{Cay}\right(S,P,h,\alpha ),S)$ of isomorphism classes of octonion algebras over R and quadratic étale algebras S;

(ii) 

The set of all pairs $\left(\mathrm{Col}\right(S,P,h,\alpha ),S)$ of isomorphism classes of colour algebras over R and quadratic étale algebras S;

(iii) 

The set of all pairs $\left(W\right(S,P,h,\alpha ),S)$ of isomorphism classes of vector algebras over R and quadratic étale algebras S.

This generalises ([3] Theorem 3.1).

Proof. 

The proof is nearly verbatim the same as the proof of the same result when R is a field, e.g., see ([3] Theorem 3.1), but we sketch a different approach here.

Fix a quadratic étale algebra S over R. Since $2\in R^{\times }$, we can write $S=\mathrm{Cay}(S,L,N)$ as a generalised Cayley–Dickson doubling of R, with L as a self-dual rank one R-module and $N:L\to R$ as a suitable nondegenerate quadratic form. Given a vector algebra $W(S,P,h,\alpha )=(P,{\times }_{\alpha })$ and S, we can define a unique octonion algebra $\left(\mathrm{Cay}\right(S,P,h,\alpha \left)\right),S)=S\oplus P$ via the multiplication

$$(a,u)(b,v)=(ab-h(v,u),va+u\overline{b}+u{\times }_{\alpha }v)$$

for $u,v\in P$ and $a,b\in S$, and a unique colour algebra $\left(\mathrm{Col}\right(S,P,h,\alpha \left)\right),S)=R\oplus P$ via

$$(a,u)(b,v)=(ab-{\displaystyle \frac{1}{2}}(h(u,v)+\overline{h(u,v)}),va+ub+u{\times }_{\alpha }v)$$

for all $u,v\in P$ and $a,b\in R$, as we have all the defining relations for their multiplication.

Conversely, the multiplication of both the algebras $\left(\mathrm{Cay}\right(S,P,h,\alpha ),S)$ and $\left(\mathrm{Col}\right(S,P,h,\alpha ),S)$ projected to P and then restricted to P yield the algebra $W(S,P,h,\alpha )=(P,{\times }_{\alpha })$ over R. Together with Proposition 3, this yields the assertion. □

Let $C=\mathrm{Col}(S,P,h,\alpha )$, $A=\mathrm{Cay}(S,P,h,\alpha )$ and define

$${\mathrm{Aut}}_S\left(A\right)=\{f\in \mathrm{Aut}\left(A\right),f{|}_S=id\},{\mathrm{Der}}_S\left(A\right)=\{d\in \mathrm{Der}\left(A\right),d{|}_S=0\}.$$

Because of the way the multiplications on A and C are defined, it is straightforward to see that $\mathrm{Der}\left(W\right(S,P,h,\alpha \left)\right)$ can be embedded into $\mathrm{Der}\left(A\right)$ and $\mathrm{Der}\left(C\right)$. Since $d\left(1\right)=0$ for every $d\in \mathrm{Der}\left(C\right)$, we have ${d|}_P\in \mathrm{Der}\left(W(S,P,h,\alpha )\right)$. Since ${d|}_S=0$ for every $d\in {\mathrm{Der}}_S\left(A\right)$, we have ${d|}_P\in \mathrm{Der}\left(W(S,P,h,\alpha )\right)$. Thus as over base fields, we obtain the following.

Lemma 2.

$\mathrm{Der}\left(\mathrm{Col}(S,P,h,\alpha )\right)={\mathrm{Der}}_S\left(\mathrm{Cay}(S,P,h,\alpha )\right)=\mathrm{Der}\left(W(S,P,h,\alpha )\right).$

Define the Lie algebras

$$u(P,h)=\{f\in {\mathrm{End}}_S\left(P\right)|h(f\left(u\right),v)+h(u,f\left(v\right))=0\mathrm{for}\mathrm{all}u,v\in P\},$$

$$su(P,h)=\{f\in u(P,h)|t{r}_S\left(f\right)=0\}.$$

When $R=F$ is a field, $\mathrm{Der}\left(W\right(S,P,h,\alpha \left)\right)=su(P,h)$, when $S=F\times F$ is split, we have hence $\mathrm{Der}\left(W\right(S,P,h,\alpha \left)\right)=sl(3,F)$ ([3] Theorem 3.4).

For $u,w\in P$ define the S-linear map

$${\lambda }_{u,v}:P\to P,x\mapsto h(x,u)v-h(x,v)u.$$

Then the R-linear span of the maps ${\lambda }_{u,v}$ with $u,v$ running through all elements in P is contained in $u(P,h)$. When R is a field then $u(P,h)$ equals this R-linear span of the maps ${\lambda }_{u,v}$.

Proposition 4.

We have

$${\mathrm{Aut}}_S\left(C\right)=\mathrm{Aut}\left(W(S,P,h,\alpha )\right)\cap {\mathrm{End}}_{S}P$$

and

$${\mathrm{Aut}}_R\left(C\right)=\mathrm{Aut}\left(W(S,P,h,\alpha )\right).$$

Proof. 

The proof is identical to the proof for base fields given in ([3], p. 1301). Firstly, $\mathrm{Aut}\left(W\right(S,P,h,\alpha \left)\right)$ can be embedded into $\mathrm{Aut}\left(C\right)$ via $f\mapsto G_f$, where we define the automorphism $G_f:C\to C$ via $G_f\left((a,u)\right)=(a,f\left(u\right))$ for all $a\in R$, $u\in P$. Put $H=\mathrm{Aut}\left(W(S,P,h,\alpha )\right)\cap {\mathrm{End}}_{S}P$. Then every $f\in H$ can be extended to $G_f\in {\mathrm{Aut}}_S\left(C\right)$ and we can embed H into ${\mathrm{Aut}}_S\left(C\right)$ via $f\mapsto G_f$. Conversely, because of the multiplicative structure of C, every $G\in {\mathrm{Aut}}_S\left(C\right)$ restricts to an automorphism ${G|}_P$ in $\mathrm{Aut}\left(W\right(S,P,h,\alpha \left)\right)$ and an endomorphism in ${\mathrm{End}}_{S}P$, and every $G\in \mathrm{Aut}\left(C\right)$ restricts to an automorphism ${G|}_P$ in $\mathrm{Aut}\left(W\right(S,P,h,\alpha \left)\right)$. This implies that ${\mathrm{Aut}}_S\left(C\right)=H$ and that $\mathrm{Aut}\left(C\right)=\mathrm{Aut}\left(W\right(S,P,h,\alpha \left)\right)$ □

Define the unitary group with respect to h,

$$U(P,h)=\{\phi \in \mathrm{GL}(P,S\left)\right|h\left(\phi \right(u),\phi (v\left)\right)=h(u,v)\mathrm{for}\mathrm{all}u,v\in P\},$$

as the set of isometries of h and the special unitary group with respect to h as

$$SU(P,h)=\{\phi \in U(P,h\left)\right|det\left(\phi \right)=1\}.$$

By Corollary 2,

$$SU(P,h)\subset \mathrm{Aut}\left(\mathrm{Col}\right(S,P,h,\alpha \left)\right)\mathrm{and}SU(P,h)\subset \mathrm{Aut}\left(\mathrm{Cay}\right(S,P,h,\alpha \left)\right).$$

When R is a field then ${\mathrm{Aut}}_S\left(C\right)=SU(P,h)$.

5. Natural Examples of Noncommutative Quadratic Jordan Algebras with Big Radicals

The theory of composition algebras over schemes was launched by Petersson [7] and later extended to large classes of Jordan and structurable algebras, e.g., [17,18,19,20,21,22]. We refer the reader to [7] for the details on how to transfer the language of nonassociative algebras over base rings to algebras over base schemes (and vice versa). Our goal is to point out how much richer the theory of colour algebras becomes when we study them over schemes, adjusting the approach of ([7] Section 3.8) in the following. To avoid pathological cases, we only look at algebras over schemes that are defined over locally free ${\mathcal{O}}_X$-modules of constant rank 7 and have full support in the sense of [7]. A colour algebra $\mathcal{C}$ over X is then defined as an algebra over X, where the stalk ${\mathcal{C}}_P$ is a colour algebra over ${\mathcal{O}}_{P,X}$ for all $P\in X$. For $P\in X$, ${\mathcal{O}}_{P,X}$ denotes the local ring of the structure sheaf ${\mathcal{O}}_X$ at P.

Let $S=R[t_0,\dots ,t_n]$ be equipped with the well-known canonical grading $S={\oplus }_{m\ge 0}{S}_m$, where $\mathrm{rank}{S}_m=\left({\displaystyle \genfrac{}{}{0pt}{}{m+n}{n}}\right)$. Let $X=\mathrm{Proj}S={\mathbb{P}}_R^n$.

For all $m\in \mathbb{Z}$, ${\mathcal{O}}_X\left(m\right)$ is a locally free ${\mathcal{O}}_X$-module of rank one, and we know that

$$H^0(X,{\mathcal{O}}_X\left(m\right))=S_m\mathrm{for}m\ge 0,H^0(X,{\mathcal{O}}_X\left(m\right))=0\mathrm{for}m<0.$$

For all $m,l\in \mathbb{Z}$, the ${\mathcal{O}}_X$-module ${\mathcal{O}}_X\left(l\right)\oplus {\mathcal{O}}_X\left(m\right)\oplus {\mathcal{O}}_X(-l-m)$ is locally free of rank three and there exists an isomorphism

$$\alpha :\det ({\mathcal{O}}_X\left(l\right)\oplus {\mathcal{O}}_X\left(m\right)\oplus {\mathcal{O}}_X(-l-m))\to {\mathcal{O}}_X.$$

For all positive integers $l,m$, we define the split colour algebra

$${\mathcal{C}}_{l,m}=\mathrm{Col}({\mathcal{O}}_X\left(l\right)\oplus {\mathcal{O}}_X\left(m\right)\oplus {\mathcal{O}}_X(-l-m),\alpha ),$$

$$\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right]\left[\begin{array}{cc}b& v\\ \check{v}& b\end{array}\right]=\left[\begin{array}{cc}ab+{\displaystyle \frac{1}{2}}(⟨u,\check{v}⟩+⟨v,\check{u}⟩)& av+bu-\check{u}\times \check{v}\\ b\check{u}+a\check{v}+u\times v& ab+{\displaystyle \frac{1}{2}}(⟨u,\check{v}⟩+⟨v,\check{u}⟩)\end{array}\right].$$

over X. The colour algebra ${\mathcal{C}}_{l,m}$ is a quadratic algebra with norm

$$n(\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right])=a^2-⟨u,\check{u}⟩.$$

The localisations of the algebras ${\mathcal{C}}_{l,m}$ at $P\in X$ are split colour algebras over ${\mathcal{O}}_{P,X}$, and the algebras ${\left({\mathcal{C}}_{l,m}\right)}_P{\otimes }_{{\mathcal{O}}_{P,X}}k\left(P\right)\cong \mathrm{Zor}\left(k\left(P\right)\right)$ are split colour algebras over the residue class fields $k\left(P\right)$, for all $P\in X$.

The global sections $H^0(X,{\mathcal{C}}_{l,m})$ of a split colour algebra ${\mathcal{C}}_{l,m}$ provide us with canonical examples of flexible quadratic algebras $C_{l,m}=H^0(X,{\mathcal{C}}_{l,m})$ over $R=H^0(X,{\mathcal{O}}_X)$: the algebra

$$C_{l,m}=\left[\begin{array}{cc}R& S_l\oplus S_m\\ S_{l+m}& R\end{array}\right]$$

is defined on the free module $R^s$ with

$$s=1+\left({\displaystyle \genfrac{}{}{0pt}{}{l+n}{n}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{m+n}{n}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{(l+m)+n}{n}}\right)$$

and is an R-subalgebra of $\mathrm{Col}\left(S\right)$ and hence is a quadratic noncommutative Jordan R-subalgebra of $\mathrm{Col}\left(S\right)$. For instance, if $n=1$ then $s=5+2(l+m)$ is always odd. Its multiplication is defined via

$$\left[\begin{array}{cc}a& f_l\oplus f_m\\ f_{l+m}& a\end{array}\right]\left[\begin{array}{cc}b& g_l\oplus g_m\\ g_{l+m}& b\end{array}\right]$$

$$=\left[\begin{array}{cc}ab& (a{g}_l+b{f}_l)\oplus (a{g}_m+b{f}_m)\\ b{f}_{l+m}+a{g}_{l+m}+f_l{g}_m-f_m{g}_l& ab\end{array}\right]$$

for $a,b\in R$, where the f’s and g’s are homogeneous polynomials in S with subscripts indicating their degrees. Note that here the terms corresponding to $⟨u,\check{v}⟩$ and $⟨v,\check{u}⟩$ vanish, $u\times v$ corresponds to $(f_l\oplus f_m)\otimes (g_l\oplus g_m)=f_l{g}_m-f_m{g}_l$ and $\check{u}\times \check{v}$ corresponds to $f_{l+m}\times g_{l+m}=0$ for all elements $f,g$.

Let n be the norm of ${\mathcal{C}}_{l,m}$, and let $n_0=n\left(X\right):H^0(X,{\mathcal{C}}_{l,m})\to H^0(X,{\mathcal{O}}_X)$. If R is a field then

$$n_0(\left[\begin{array}{cc}a& u\\ \check{u}& a\end{array}\right])=a^2$$

is degenerate, and $C_{l,m}$ is a flexible quadratic algebra with the radical

$$\mathrm{rad}{C}_{l,m}=\mathrm{rad}{n}_0=\left[\begin{array}{cc}0& S_l\oplus S_m\\ S_{l+m}& 0\end{array}\right]$$

of dimension $1+\left({\displaystyle \genfrac{}{}{0pt}{}{l+n}{n}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{m+n}{n}}\right)+\left({\displaystyle \genfrac{}{}{0pt}{}{(l+m)+n}{n}}\right)$. Analogously as observed in ([7] Section 3.8), when R is a field, we see that the radical of $C_{l,m}$ is nilpotent:

$${\left(\mathrm{rad}{C}_{l,m}\right)}^2=\left[\begin{array}{cc}0& 0\\ S_{l+m}& 0\end{array}\right]\mathrm{and}{\left(\mathrm{rad}{C}_{l,m}\right)}^2=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right].$$

Noncommutative quadratic Jordan algebras form a huge class of algebras. The algebras we constructed here are explicit examples which to our knowledge have not appeared anywhere in the literature yet. They stand out, as they can be all seen as being embedded in the colour algebra $\mathrm{Col}\left(R[t_0,\dots ,t_n]\right)$ as R-subalgebras, and they can also be directly constructed out of the quadratic alternative algebras obtained in an analogous construction by Petersson ([7] Section 3.8), which in turn all can be embedded in the Zorn algebra $\mathrm{Zor}\left(R[t_0,\dots ,t_n]\right)$ as R-subalgebras. For general constructions of flexible quadratic algebras over rings and a comprehensive overview, see ref. [14].

Remark 4.

The structure of the quadratic algebras $C_{l,m}$ is of the type denoted by $\widehat{A}=(R,F,B,\times )$ in ([14] Example 2), with $B(u,v)={\displaystyle \frac{1}{2}}(h(u,v)+\overline{h(u,v)})$, where $A=(S,F,h,\times )$ is an algebra with a scalar involution, S is a quadratic étale algebra, and h is degenerate.

In particular, if $A=\mathrm{Cay}(T,F,h,{\times }_{\alpha })$ is an octonion algebra over a field R of characteristic not 2, then $\widehat{A}=(R,F,B,{\times }_{\alpha })$ is the colour algebra. So we could view these algebras as “degenerate” generalisations of a colour algebra.

6. Examples of Non-Split Colour Algebras over Curves of Genus One

Explicit examples of octonion algebras over curves X of genus one with an interesting ${\mathcal{O}}_X$-module structure were constructed in [23,24] by employing the generalised Cayley–Dickson doubling process over schemes (resp. rings) introduced by Petersson in ref. ([7] Section 2.5).

Interesting examples of colour algebras can easily be obtained by using the “non-split” octonion algebras presented in ([23] Proposition 2.11 (a)) (i.e., ocotonion algebras not isomorphic to $\mathrm{Zor}(\mathcal{T},\alpha$). We will only briefly sketch the approach here. For unexplained terminology or results used in this section, we refer the reader to refs. [23,24].

Let k be a perfect field of characteristic not equal to two and let X be a smooth projective curve of genus one and arbitrary index over k. Then we know that ${}_2\mathrm{Pic}\left(X\right)=\left\{{\mathcal{O}}_X\right\}, {}_2\mathrm{Pic}\left(\mathrm{X}\right)=\{{\mathcal{O}}_{\mathrm{X}},{\mathcal{L}}_1\} \mathrm{or} {}_2\mathrm{Pic}\left(\mathrm{X}\right)=\{{\mathcal{O}}_{\mathrm{X}},{\mathcal{L}}_1,{\mathcal{L}}_2,{\mathcal{L}}_3\}\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_2.$

Let ${\mathcal{F}}_2$ be the absolutely irreducible ${\mathcal{O}}_X$-module of rank two with a trivial determinant which is uniquely determined up to isomorphism.

We will look at Cayley–Dickson doublings of the quaternion algebra $\mathcal{D}=\mathcal{E}n{d}_X\left({\mathcal{F}}_2\right)$ over ${\mathcal{O}}_X$; note that the norm of $\mathcal{D}$ is given by the determinant. It is well known that the ${\mathcal{O}}_X$-module structure of $\mathcal{D}$ is given by $\mathcal{D}\cong {\mathcal{O}}_X\oplus {\mathcal{F}}_3$, where ${\mathcal{F}}_3$ is the absolutely indecomposable self-dual vector bundle on X of rank three with a trivial determinant, which means $det{\mathcal{F}}_3\cong {\mathcal{O}}_X$. Because the Theorem of Krull–Schmidt holds in our setting, this module structure shows that this quaternion algebra only has ${\mathcal{O}}_X$ as a composition subalgebra.

Now define a locally free projective ${\mathcal{O}}_X$-module of constant rank four via

$$\mathcal{P}=\mathcal{H}o{m}_X({\mathcal{F}}_2,{\mathcal{L}}_i\otimes {\mathcal{F}}_2)\cong {\mathcal{L}}_i\oplus {\mathcal{L}}_i\otimes {\mathcal{F}}_3,$$

where ${\mathcal{L}}_i{\in }_2\mathrm{Pic}\left(X\right)$ is any nontrivial self-dual line bundle. Put $N\cong {\alpha }_i\oplus {\alpha }_i\otimes \beta$, where $\beta :{\mathcal{F}}_3\to {\mathcal{O}}_X$ is a nondegenerate quadratic form satisfying $det{\beta }_{\xi }\equiv 1mod{K}^{\times 2}$. (Here, we compute the stalk at $\xi \in X^{\prime }$, the generic point of the Jacobian $X^{\prime }$ of X.) Then the generalised Cayley–Dickson doubling $\mathcal{A}=\mathrm{Cay}(\mathcal{D},\mathcal{P},aN)$ of $\mathcal{D}$, $a\in k^{\times }$, is an octonion algebra over ${\mathcal{O}}_X$ ([23] Proposition 2.11 (a)).

The ${\mathcal{O}}_X$-module structure of $\mathcal{A}$ is given by

$${\mathcal{O}}_X\oplus {\mathcal{F}}_3\oplus {\mathcal{L}}_i\oplus {\mathcal{L}}_i\otimes {\mathcal{F}}_3.$$

Again by the Theorem of Krull–Schmidt, this module structure implies that the octonion algebra $\mathcal{A}$ cannot have a split torus (i.e., a split composition subalgebra of rank 2 isomorphic to ${\mathcal{O}}_X\times {\mathcal{O}}_X$) as a subalgebra, i.e., a non-split composition subalgebra of rank 2. Hence $\mathcal{A}$ is not isomorphic to a Zorn algebra over ${\mathcal{O}}_X$. A routine argument shows that $\mathcal{A}$ does contain a composition subalgebra of rank 2 over ${\mathcal{O}}_X$ of the type $\mathcal{S}=\mathrm{Cay}({\mathcal{O}}_X,{\mathcal{L}}_i,N_0)$, with $N_0{=⟨1⟩\perp aN|}_{{\mathcal{L}}_i}$. Therefore, $\mathcal{A}$ can be expressed employing hermitian forms, analogously as over rings, and can be written as a colour algebra of the type

$$\mathrm{Col}(\mathcal{S},{\mathcal{F}}_3\oplus {\mathcal{L}}_i\oplus {\mathcal{L}}_i\otimes {\mathcal{F}}_3,h,\alpha )$$

for a suitably globally defined hermitian form h that can be derived from N and some $\alpha$.

This colour algebra has the ${\mathcal{O}}_X$-module structure

$${\mathcal{O}}_X\oplus {\mathcal{F}}_3\oplus {\mathcal{L}}_i\otimes {\mathcal{F}}_3.$$

By the Theorem of Krull–Schmidt, this colour algebra cannot be isomorphic to a split colour algebra since we know that the dual of ${\mathcal{L}}_i\otimes {\mathcal{F}}_3$ is not ${\mathcal{F}}_3.$

7. Discussion

This exposition is only a first step towards understanding colour algebras over rings, as we focused exclusively on colour algebras constructed via hermitian forms. It is not clear if the colour algebra constructions we presented here are the only possible ones.

We propose the following open questions for further research, some of which have been asked by one of the referees:

• For octonion algebras over rings, it can happen that two non-isomorphic octonion algebras have isometric norms [25]. Can this also happen for colour algebras?
• Over a local ring where two is invertible, every octonion algebra can be constructed using a suitable hermitian form over one of its quadratic étale algebras. Is there any obstruction for a global construction of a colour algebra from a hermitian form, and if so, what is the precise obstruction, given local feasibility? And are there reasonable additional assumptions on R under which every colour algebra does arise globally from such a hermitian form?
• Is it possible to construct colour algebras without employing hermitian forms? This question is directly linked to the problem if it is possible to construct a colour algebra on an indecomposable R-module of rank 7.
• How do the automorphism groups differ concretely between colour algebras over rings versus fields, or are they the same?
• Is there any physical interpretation of moving the quark model’s colour symmetry from $Col\left(\mathbb{C}\right)$ to a ring R?
• The split octonions and are directly related to Penrose’s classical twistors. Can this relation be better explained by using colour algebras?
• What is the most promising next research direction opened by our results?

8. Materials and Methods

All methods used have been explained and cited in this publication.

No large datasets have been used. No AI has been used.