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We study real algebras admitting reflections which commute. In dimension two, we show that two commuting reflections coincide and we specify the two and four-dimensional real algebras cases. We characterize real algebras of division of two-dimensional to third power-associative having a reflection. Finally We give a characterization in four-dimensional, the unitary real algebras of division at third power-associative having two reflections that commute. In eight-dimensional, we give an example of algebra so the group of automorphisms contains a subgroup isomorphic to \(\mathbb{Z}_2\times\mathbb{Z}_2\).
The classification of non-associative division algebras over a commutative field \(\mathbb{K}\) with a characteristic different from \(2\) is a pastionning and topical problem, whose origins date back to the discovery of quaternions (\(\mathbb{H}\), Hamilton 1843) and octonions (\(\mathbb{O}\), Graves 1843, Cayley 1845). Fundamental results appeared, Hopf proved that the dimension of a real algebra of division of finite dimension \(n\) is a power of 2 and cannot exceed 2 in the commutative case.
Bott and Milnor [1] refined the result of Hopf by reducing the power of \(2\) to \(n\in \{1,2, 4, 8\}\). It is trivial to show that in dimension one the real algebra \(\mathbb{R}\) is unique. In two-dimensional, the classification of these algebras is recently completed. The problem remains open in dimension 4 and 8. The study is quite interesting when there is a sufficient number of distinct reflections which commute for a finite-dimensional division algebra since the product of this last translates a certain symmetry and elegance.
In this paper, we give a description of real algebras of division in two-dimensional having one reflection, and in four-dimensional having two distinct reflections which commute. We recall that the subgroup generated by the latter is isomorphic to the Klein’s group \(\mathbb{Z}_2\times \mathbb{Z}_2\) with \(\displaystyle\mathbb{Z}_2:=\frac{\mathbb{Z}}{2\mathbb{Z}}\).
In eight-dimensional, we give a class of algebra whose group of automorphisms contains the Klein’s group without giving the necessary and sufficient condition of the division.
Let \(A\) be an arbitrary non-associative real algebra. We define, \(I(A)=\{x\in A, x^2=x\}\) and let \(x\), \(y\in A\), \([x,y]=xy-yx\) and \((x,y,z)=(xy)z-x(yz)\).
\(A\) is said to be
– division if the operations \(L_x: A\rightarrow A\), \(y\mapsto xy\) and \(R_x: A\rightarrow A\), \(y\mapsto yx\) are bijective, for all \(x\in A\), \(x\neq 0.\)
–at third power-associative if \((x,x,x)=0\) for all \(x\in A\).
–at 121 power-associative if \((x,x^2,x)=0\) for all \(x\in A\).
A linear map \(\partial: A\rightarrow A\) is said to be a derivation of \(A\) if for all \(x,\) \(y\in A\), we have \(\partial(x.y)=\partial(x).y+x.\partial(y)\). The derivations of \(A\) form a vector subspace of the endomorphisms of \(A\) (\(End_\mathbb{K}(A)\)) which is a Lie algebra, for the bracket of Lie \([f,g]=f\circ g- g\circ f\). Such an algebra is called the Lie algebra of the derivations of \(A\) denoted by \(Der(A).\)
A linear map \(f: A\rightarrow A\) is said to be an automorphism of \(A\) if \(f\) is bijective and for all \(x,\) \(y\in A\), \(f(x.y)=f(x).f(y).\) The automorphisms of \(A\) constitute a group \(Aut_\mathbb{K}(A)\) for the usual law. \(f\in Aut(A)\) is said to be a reflection of \(A,\) if \(f\) is involutive \((f\circ f=id_A)\) not identical \((f\neq id_A)\). Let \(f\) be an automorphism of \(A\) and \(\lambda \in \mathbb{R}\), we denote by \(E_{\lambda}(f)\) the kernel of \(f-\lambda id_A\), \(id_A\) being the identity operator of \(A\).
Let \(f, g: A\rightarrow A\) be linear bijections of \(A\). Recall that the \((f,g)\)-Albert isotope of \(A\) denoted by \(A_{f,g}\) is a vector space \(A\) with the product \(x\odot y = f(x)g(y)\).
Remark 1. A linearization of \((x,x,x)=0\), gives \([x^2,y]+[xy+yx,x]=0\) for all \(x\), \(y\in A\). So by taking \(y=x^2\) we get \([x.x^2+x^2.x, x]=0\Rightarrow 2[x.x^2,x]=0\Rightarrow (x.x^2)x-x(x.x^2)=0\Rightarrow (x.x^2)x-x(x^2.x)=0\Rightarrow (x,x^2,x)=0\). We can therefore affirm that if \(A\) is at third power-associative then it is at 121 power-associative. But the converse is not true in the case where the algebra is not of division. For example the algebra \(A\) having a basis \(\{e_1, e_2, e_3, e_4\}\) whose product of the elements in this base is given by, \({e_1}^2=e_1\), \(e_1e_2=e_3\), \(e_2e_1=e_4\) and other null products, is at 121 power-associative and it is not at third power-associative.
In [2], we have the following result;
Lemma 1. Let \(A\) be a real algebra of division of finite 2n-dimensional with \(n\in\{1,2,4\}\). We suppose that there exists an automorphism \(f\) of \(A\) such that \(sp(f)=\{-1,1\}\). Then the following inclusions between subalgebras of \(A\) are strict, \[\{0\}\subset E_1(f)\subset E_1(f)+E_{-1}(f).\] In that case \(dim E_1(f)=n\geq 1,\) and \(dim \big( E_1(f)+E_{-1}(f)\big)=2n.\)
In addition, the following statements are equivalent,
And the following equalities hold, \[E_1(f)E_{-1}(f)=E_{-1}(f)E_1(f)=E_{-1}(f),\] \[E_{-1}(f).E_{-1}(f)=E_1(f).\]
Definition 1. Let \(\alpha\), \(\beta,\) \(\gamma,\) \(\alpha',\) \(\beta'\) and \(\gamma'\) be real numbers. We define
algebra \(A\) having a base \(\{e,e_1, e_2, e_3\}\) for which the product
is given as
\[
\label{tab1}\tag{1}
\begin{array}{c|c|c|c|c}
\odot & e & e_1 & e_2 & e_3\\
\hline
e & e & e_1 & e_2 & e_3\\
\hline
e_1 & e_1 & -e & \gamma e_3 & \beta’ e_2\\
\hline
e_2 & e_2 & \gamma’ e_3 & -e & \alpha e_1\\
\hline
e_3 & e_3 & \beta e_2 & \alpha’ e_1 & -e\\
\end{array}
\]
We denote this algebra \(A\) by \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\).
In [3], the author gives the following results:
Theorem 1. A necessary condition that the algebra \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\) should be a division algebra, is that the six nonzero constants \(\alpha,\) \(\beta,\) \(\gamma,\) \(-\alpha',\) \(-\beta',\) \(-\gamma'\) should have the same sign. This sign may be taken to be positive without loss of generality. If the six constants are positive, a sufficient (but not necessary) condition for a division algebra is that they satisfy the relation \(f(\alpha, \beta, \gamma)=f(-\alpha', -\beta', -\gamma')\) where \(f\) is defined by \[f(x,y,z)=x+y+z-xyz.\]
Theorem 2. A necessary and sufficient condition that the division algebra \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\) should have two sided rank 2 is that the equations \(\alpha'=\alpha\), \(\beta'=\beta\) and \(\gamma'=\gamma\) hold true. In the contrary case, the algebra has two sided rank 4.
Proposition 1. Let \(A\) be a finite dimensional real algebra. The following proposals are equivalent,
\(Aut(A)\) contains two distinct reflections which commute.
\(Aut(A)\) contains a subgroup isomorphic to the Klein’s group \(\mathbb{Z}_2\times \mathbb{Z}_2\).
Proof. \((1)\Longrightarrow (2)\) Let \(A\) be a real algebra such that \(Aut(A)\) contains two reflections which commute \(f\) and \(g.\) Then \(h=f\circ g \in Aut(A)\) and is a different reflection of \(f\) and \(g\). The subgroup of \(Aut(A)\) generated by \(f\) and \(g\) is isomorphic to the klein’s group \(\mathbb{Z}_2\times \mathbb{Z}_2\).
\((2)\Longrightarrow (1)\) Obvious because the elements of the subgroup of \(Aut(A)\) which is isomorphic to \(\mathbb{Z}_2\times \mathbb{Z}_2\) different from the identity are distinct reflections that commute. ◻
Proposition 2. Let \(A\) be a two-dimensional division algebra and let \(f\), \(g\) be two commuting reflections of \(A\). Then \(f=g\)
Proof. The endomorphisms \(f\) and \(g\) of \(A\) are diagonalizable which commute, consequently there exists a common basis \(\{e, e_1\}\) of \(A\), formed of eigenvectors associated with eigenvalues \(1\) and \(-1\). According to Lemma 1. The eigpaces \(E_1(f)\), \(E_{-1}(f)\), \(E_1(g)\), \(E_{-1}(g)\) of \(f\) and \(g\) are one-dimensional. By setting \(E_1(f)= \mathbb{R}e\) and \(E_{-1}(f)=\mathbb{R}e_1\).
Suppose that \(E_1(g)=\mathbb{R}e_1\) and \(E_{-1}(g)=\mathbb{R}e\), let \(x=a e+b e_1\in A,\) we have \[f(x)=af(e)+bf(e_1)=ae-be_1=-ag(e)-bg(e_1)=-g(ae+be_1)=-g(x),\] thus \(f=-g\) (which is not an automophism of \(A\)), absurd. So \(E_1(g)=\mathbb{R}e\) and \(E_{-1}(g)=\mathbb{R}e_1\), therefore \(f\) and \(g\) coincide on \(A=E_1(f)\oplus E_{-1}(f).\) ◻
Theorem 3. Let \(A\) be a two-dimensional division algebra
and let \(f\) be a reflections of \(A.\) Then there exists a basis \(B=\{e, e_1\}\) of \(A\) such that the product of \(A\) in this basis is given as,
\[
\label{tab2}\tag{2}
\begin{array}{c|c|c}
\odot & e & e_1 \\
\hline
e & e & \alpha e_1 \\
\hline
e_1 & \beta e_1 & \gamma e \\
\end{array}
\]
with \(\alpha\),
\(\beta\), \(\gamma\) are non-zero real numbers and
\(\alpha\beta\gamma< 0\). We will
denote this algebra \(A(\alpha, \beta,
\gamma)\).
Proof. The eigenspaces \(E_1(f)\), \(E_{-1}(f)\) of f are one-dimensional and \(A=E_1(f) \oplus E_{-1}(f)\). There exists \(e \in E_1(f)\) and \(e_1 \in E_{-1}(f)\), such that \(\{e, e_1\}\) is a basis of \(A.\) Taking into account the equalities \((\ast)\) of Lemma 1, we obtain the product of the elements of this basis of (2). ◻
For the division, the Theorem 3 of [4] gives the result.
Corollary 1. Let \(A\) be a two-dimensional division algebra, then the following propositions are equivalent,
\(Aut(A)\) contains a reflection.
\(A\) is isomorphic to \(A(\alpha, \beta, \gamma)\) with \(\alpha\beta\gamma< 0\).
Proof. \((1)\Rightarrow (2)\) The Theorem 3 gives the result.
\((2) \Rightarrow (1)\) The endomorphism \(f\) of \(A(\alpha, \beta, \gamma)\) defined by \(f(x_0e+x_1e_1)=x_0e-x_1e_1\) is a reflection of \(A(\alpha, \beta, \gamma)\). ◻
Theorem 4. Let \(A\) be a two-dimensional division algebra and let \(f\) be a reflection of \(A\), then the following propositions are equivalent,
\(A\) is commutative.
\(A\) is isomorphic to \(A(\alpha, \beta, \gamma)\) with \(\beta=\alpha\) and \(\gamma< 0\).
\(A\) is at third power-associative.
\(A\) is at 121 power-associative.
Proof. \((1)\Rightarrow (2)\) Since \(f\) be a reflection of \(A\), the Corollary 1 asserts that \(A\) is isomorphic to \(A(\alpha, \beta, \gamma)\) with \(\alpha \beta \gamma< 0\). Since \(A\) is commutative, \(e_1e=ee_1\Longrightarrow \alpha=\beta\), therefore we get the result.
\((2)\Rightarrow (3)\) It is easy to show that \(A(\alpha, \beta, \gamma)\), with \(\beta=\alpha\) and \(\gamma< 0\), is at third power-associative.
\((3)\Rightarrow (4)\) The Remark 1 gives the result.
\((4)\Rightarrow (1)\) \(A\) is isomorphic to \(A(\alpha, \beta, \gamma)\) with \(\alpha \beta \gamma< 0\) and it is at \(121\) power-associative, We have \((e_1, {e_1}^2, e_1)=0\Rightarrow \gamma^2(\alpha-\beta)e=0\Rightarrow \alpha=\beta.\) and \(\gamma< 0.\) Then \(A\) is commutative. ◻
Theorem 5. Let \(A\) be a two-dimensional division algebra and let \(f\) be a reflection of \(A\), then A is isomorphic to \(A(\alpha, \beta, \gamma)\) and we have.
| \(I(A)\) | |
|---|---|
| \(\alpha+\beta\neq 0 \ et \ \gamma(\alpha+\beta-1)> 0\) | \(\{e, \ \ \lambda_0e-\lambda_1 e_1, \ \ \lambda_0e+\lambda_1 e_1 \}\) |
| otherwise | \(\{e\}\) |
with \(\displaystyle \lambda_0=\frac{1}{\alpha+\beta}\), \(\displaystyle {\lambda_1}^2=\frac{\alpha+\beta-1}{\gamma(\alpha+\beta)^2}\).
Proof. Since \(A\) is isomorphic to \(A(\alpha, \beta, \gamma)\) with \(\alpha\beta\gamma< 0.\) Let \(x=\lambda_0 e+\lambda_1e_1 \in I(A)\), we have \(x^2=x\Leftrightarrow \left\{\begin{array}{ll} {\lambda_0}^2+\gamma{\lambda_1}^2= \lambda_0;\\ (\alpha+\beta)\lambda_0\lambda_1=\lambda_1, \end{array}\right.\) we obtain \(I(A)\) by resolving the system and discussing on \(\alpha+\beta\) and \(\displaystyle \frac{\alpha+\beta-1}{\gamma}\). ◻
Corollary 2. Let \(A\) be a two-dimensional division algebra and let \(f\) be a reflection of \(A\), then A is isomorphic to \(A(\alpha, \beta, \gamma)\) and we have
| \(Aut(A)\) is isomorphic to | |
|---|---|
| \(\alpha+\beta\neq 0 \ et \ \gamma(\alpha+\beta-1)> 0\) | \(\mathbb{Z}_2\) or \(S_3\) |
| \(\mathbb{Z}_2\) |
Proposition 3. Let \(A\) be a four-dimensional division algebra, let \(f\) and \(g\) are reflections of \(A.\) Then there exists a basis \(B=\{e, e_1, e_2, e_3\}\) of \(A\) where \(e^2=e\) and \(E_1(f)=\mathbb{R}e+\mathbb{R}e_1,\) \(E_{-1}(f)=\mathbb{R}e_2+\mathbb{R}e_3,\) \(E_1(g)=\mathbb{R}e+\mathbb{R}e_2,\) and \(E_{-1}(g)=\mathbb{R}e_1+\mathbb{R}e_3.\)
Proof. Since \(f\) and \(g\) are diagonalizable and commute, so there is a common basis \(\{e,e_1,e_2, e_3\}\) formed by eigenvectors associated to eigenvalues \(1\) or \(-1\). The Lemma 1 shows that \(E_1(f)\), \(E_{-1}(f)\), \(E_1(g)\) and \(E_{-1}(g)\) are vector spaces of two-dimensional. If \(E_1(f)=\mathbb{R}e+\mathbb{R}e_1\), \(E_{-1}(f)=\mathbb{R}e_2+\mathbb{R}e_3\), then one of the eigenvectors \(e\), \(e_1\) (and only one) belongs to \(E_{1}(g)\), otherwise \(f\) and \(g\) coincide in \(A=E_1(f)\oplus E_{-1}(f)\).
We can therefore set \(E_1(g)=\mathbb{R}e+\mathbb{R}e_2\) and \(E_{-1}(g)=\mathbb{R}e_1+\mathbb{R}e_2\). We have \(e_1\in E_1(f)\Rightarrow {e_1}^2\in E_1(f).E_1(f)=E_1(f),\) and \(e_1\in E_{-1}(g)\Rightarrow {e_1}^2\in E_{-1}(g).E_{-1}(g)=E_1(g).\) Therefore \(e_1\in E_1(f)\cap E_1(g)=\mathbb{R}e\). Thus by analogy, the elements \({e_i}^2\in E_1(f)\cap E_1(g)=\mathbb{R}e\) for all \(i\in \{2, 3\}\). Since \(E_1(f)\cap E_1(g)\) is a subalgebra of \(A\), the element \(e\) is a scalar multiple of an idempotent and can be assumed to be idempotent. ◻
Proposition 4. Let \(A\) be a real division algebra of unit four-dimensional of unit \(e\). Then the following propositions are equivalent,
\(Aut(A)\) contains two distinct reflections which commute.
\(Aut(A)\) contains a subgroup isomorphic to the Klein’s group \(\mathbb{Z}_2\times \mathbb{Z}_2\).
\(A\) is isomorphic to the algebra \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\).
Proof. \((1) \Longleftrightarrow (2)\) This is true according to Proposition 1.
\((2) \Longleftrightarrow (3)\) We have two distinct reflections which commute. Proposition 2 ensures the existence of a basis \(\{e, e_1, e_2, e_3\}\) of \(A\), and the subalgebras \(\mathbb{R}e+\mathbb{R}e_i\), for all \(i\in \{1,2,3\}\), are isomorphic to the algebra \(\mathbb{C}\) since they are real algebras of unit division of two-dimensional. We can now set \(E_1(f)=\mathbb{R}e+\mathbb{R}e_1,\) \(E_{-1}(f)=\mathbb{R}e_2+\mathbb{R}e_3,\) \(E_1(g)=\mathbb{R}e+\mathbb{R}e_2\) and \(E_{-1}(g)=\mathbb{R}e_1+\mathbb{R}e_3.\)
We have \(e_1\in E_1(f) \ et \ e_2 \in E_{-1}(f)\Rightarrow e_1e_2, \ \ e_2e_1\in E_1(f).E_{-1}(f)=E_{-1}(f)\), \(e_1 \in E_{-1}(g)\) and \(e_2 \in E_1(g)\Rightarrow e_1e_2, \ \ e_2e_1 \in E_{-1}(g).E_1(g)=E_{-1}(g)\).
As a result \(e_1e_2,\) and \(e_2e_1 \in E_{-1}(f)\cap E_{-1}(g)=\mathbb{R}e_3\). In the same way we have \[\begin{aligned} e_1e_3, && e_3 e_1 \in E_{-1}(f)\cap E_{1}(g)=\mathbb{R}e_2, \\ e_2e_3, && e_3 e_2 \in E_{1}(f)\cap E_{-1}(g)=\mathbb{R}e_1. \end{aligned}\] This gives the multiplication table for (1).
\((3)\Longrightarrow (1)\) \(f: A\longrightarrow A;\) \(\lambda_0 e+\sum\limits_{i=1}^{3}\lambda_i e_i\longmapsto \lambda_0 e+\lambda_1 e_1-\sum\limits_{i=2}^{3}\lambda_i e_i\) and \(g: A\longrightarrow A;\) \(\lambda_0 e+\sum\limits_{i=1}^{3}\lambda_i e_i\longmapsto \lambda_0 e-\lambda_1 e_1+\lambda_2 e_2-\lambda_3e_3\) are distinct reflections that commute. ◻
Corollary 3. If \(A\) is a real division algebra of four-dimensional with two commute distinct reflections then \(A\) is isotope in the sens of Albert to \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\) with \(e\in I(A)\).
Proof. Let \(A\) be a real division algebra of four-dimensional having two distinct reflections that commute \(f\) and \(g\). According to Proposition 2 there is a basis \(\{e, e_1, e_2, e_3\}\) where \(e\in I(A)\) and \(f(e)=g(e)=e\). Let \(x \in A\), we have \[f\circ R_e(x)=f(R_e(x))=f(xe)=f(x)f(e)=f(x)e=R_e(f(x))=R_e\circ f(x),\] so \[f\circ R_e=R_e\circ f\Leftrightarrow f\circ {R_e}^{-1}={R_e}^{-1}\circ f.\] As well as \(f\circ {L_e}^{-1}={L_e}^{-1}\circ f\), \((A, \odot)\) where \(x\odot y={R_e}^{-1}(x).{L_e}^{-1}(y)\) for all \(x\), \(y \in A\), is isotope to \(A\), and is unitary of unit \(e\). We have, \[f(x\odot y)=f\big({R_e}^{-1}(x).{L_e}^{-1}(y)\big)=f\big({R_e}^{-1}(x)\big).f\big({L_e}^{-1}(y)\big)={R_e}^{-1}(f(x)).{L_e}^{-1}(f(y))=f(x)\odot f(y),\] then \(f\in Aut((A,\odot))\). We also show by analogy that \(g\in Aut((A,\odot)).\) Hence \((A,\odot)\) verifies the assumptions of Proposition 5 therefore it is isomorphic to \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\). ◻
Theorem 6. Let \(A\) be a real division algebra of unit four-dimensional having two reflections that commute. Then the following statements are equivalent,
\(A\) is isomorphic to \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\), with \(\alpha'=\alpha\), \(\beta'=\beta\), \(\gamma'=\gamma\).
\(A\) is at third power-associative.
\(A\) is at 121 power-associative.
Proof. \((1)\Longrightarrow(2)\) For all \(x=\displaystyle \lambda_0e+\sum\limits_{i=1}^{3}\lambda_ie_i\in\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\), with \(\alpha'=\alpha\), \(\beta'=\beta\), \(\gamma'=\gamma\), we have \(x^2=-N(x)e+2\lambda_0 x\) with \(N(x)=\displaystyle\sum\limits_{i=0}^{3}{\lambda_i}^2\) and it is easy to check that \((x,x,x)=0\)
\((2)\Longrightarrow(3)\) It is clear from Remark 1
\((3)\Longrightarrow(1)\) A simple calculation show that \(\alpha=\alpha'\), \(\beta=\beta'\) et \(\gamma=\gamma'\). ◻
We notice that \(\{1,2,3\} := J_3;\) \(\Big\{\sum_{i=1}^{3}\alpha_ie_i, \ \alpha_i\in \mathbb{R} \Big\}:= V;\) \(\{x\in V; x^2=-e\}:= V_e\) and \(\{\lambda e_i+\mu e_j; \ \lambda^2+\mu^2=1\} := V_{ij} \ \ \ i \ j \in J_3, \ i\neq j.\)
Proposition 5. Let \(D=\{x\in\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma'); x^2=-e \}\). If \((\alpha-\alpha')(\beta-\beta')(\gamma-\gamma')=0\), then \(D=\Big\{\lambda_1e_1+\lambda_2e_2+\lambda_3e_3; \displaystyle \sum_{i=1}^{3}{\lambda_i}^2=1\Big\}\), otherwise, \(D=\{-e_1,e_1, -e_2, e_2, -e_3, e_3\}\)
Proof. Let \(x=\lambda_0 e+\displaystyle\sum_{i=1}^{3}\lambda_ie_i\) and \(x'=\lambda_0' e+\displaystyle\sum_{i=1}^{3}\lambda_i'e_i \in\) \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\), we have \(xx'=\big(\lambda_0\lambda_0'-\lambda_1\lambda_1'-\lambda_2\lambda_2'-\lambda_3\lambda_3'\big)e +\big(\lambda_0\lambda_1'+\lambda_0'\lambda_1+\alpha\lambda_2\lambda_3'-\alpha'\lambda_3\lambda_2'\big)e_1+\big(\lambda_0\lambda_2'+\lambda_0'\lambda_2+\beta\lambda_3\lambda_1'-\beta'\lambda_1\lambda_3'\big)e_2+ \big(\lambda_0\lambda_3'+\lambda_0'\lambda_3+\gamma\lambda_1\lambda_2'-\gamma'\lambda_2\lambda_1'\big)e_3.\) Especially \(x^2=\big({\lambda_0}^2-{\lambda_1}^2-{\lambda_2}^2-{\lambda_3}^2\big)e +\big(2\lambda_0\lambda_1+(\alpha-\alpha')\lambda_2\lambda_3\big)e_1+\big(2\lambda_0\lambda_2+(\beta-\beta')\lambda_1\lambda_3\big)e_2+ \big(2\lambda_0\lambda_3+(\gamma-\gamma')\lambda_1\lambda_2\big)e_3.\) So \[x\in D\Longleftrightarrow \left\{\begin{array}{ll} {\lambda_0}^2-{\lambda_1}^2-{\lambda_2}^2-{\lambda_3}^2=-1;\\ 2\lambda_0\lambda_1+(\alpha-\alpha')\lambda_2\lambda_3 =0;\\ 2\lambda_0\lambda_2+(\beta-\beta')\lambda_1\lambda_3=0; \\ 2\lambda_0\lambda_3+(\gamma-\gamma')\lambda_1\lambda_2=0, \end{array}\right.\] we obtain \(D\) by resolving the system and discussing on \((\alpha-\alpha')(\beta-\beta')(\gamma-\gamma')=0\) and \((\alpha-\alpha')(\beta-\beta')(\gamma-\gamma')\neq0\). ◻
Proposition 6. The group of automorphisms of algebra \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\) contains two reflections which commute. Furthermore, the following two propositions are equivalent:
\(\alpha=\alpha'=\beta=\beta'=\gamma=\gamma'\).
\(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\) is isomorphic to \(\mathbb{H}^{(\frac{\alpha+1}{2})}.\)
Proof. Proposition 4 shows that \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\) contains two reflections which commute. ◻
Lemma 2. Let \(f\in Aut(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma'))\), we have
\(f(e)=e\),
\(f(e_i)^2=-e\) for all \(i\in J_3\),
\(\left\{\begin{array}{ll} \gamma'f(e_1)f(e_2)+\gamma f(e_2)f(e_1)=0, \ \ \ \ \ E(3.1)\\ \beta f(e_1)f(e_3)+\beta' f(e_3)f(e_1)=0, \ \ \ \ \ E(3.2)\\ \alpha'f(e_2)f(e_3)+\alpha f(e_3)f(e_2)=0, \ \ \ \ \ E(3.3) \end{array}\right.\)
Lemma 3. Let \(f\in Aut(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma'))\), then \(f(V)\subseteq V.\)
Proof. The demonstration is reduced to the case \((\alpha-\alpha')(\beta-\beta')(\gamma-\gamma')\neq 0\) (Proposition 5, Lemma 2). We will show that \(\{f(e_1), f(e_2), f(e_3)\}\subseteq V\).
Let \(f(e_i)=\lambda_{0i}e+\lambda_{1i}e_1+\lambda_{2i}e_2+\lambda_{3i}e_3\) or \(i\in J_3.\) Now, \(f(e_2)f(e_3)\) and \(f(e_3)f(e_2)\) have the same component following \(e\), namely \(\lambda_{02}\lambda_{03}-\lambda_{12}\lambda_{13}-\lambda_{22}\lambda_{23}-\lambda_{23}\lambda_{33}:=\lambda\).
If \(f(e_1)\notin V\), since \(\alpha f(e_1)=f(e_2)f(e_3)\notin V\), then \(\lambda \neq 0\). E(3.3) shows that \(\alpha-\alpha'=0\) absurd. Therefore \(f(e_1)\in V\) it is the same for \(f(e_2)\) and \(f(e_3)\in V.\) ◻
Proposotion 7. We suppose that \((\alpha-\alpha')(\beta-\beta')(\gamma-\gamma')\neq 0\). Let \(f\in Aut(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma'))\). Then there exists a permutation \(\sigma\) of the set \(J_3\), which is not a transposition, and scalars \(\varepsilon_1,\) \(\varepsilon_2\in \{1,-1\}\) such that \(\Big(f(e_1), f(e_2), f(e_3)\Big)=\Big(\varepsilon_1e_{\sigma(1)}, \varepsilon_2e_{\sigma(2)}, \varepsilon_3e_{\sigma(3)} \Big)\). Furthermore, the following two are equivalent:
\(Aut(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma'))\) contains an element \(f\) which transitively permutes, \(\pm e_1\), \(\pm e_2\) and \(\pm e_3\).
\(\alpha=\beta=\gamma \neq \alpha'=\beta'=\gamma'\).
Proposotion 8. We have
\(V_e=\left\{\begin{array}{ll} \{\pm e_1, \pm e_2, \pm e_3 \}, \ \ \ if \ \ (\alpha-\alpha')(\beta-\beta')(\gamma-\gamma')\neq 0,\\ \{\pm e_1\}\cup V_{23}, \ \ \ \ if \ \ \alpha=\alpha' \ and \ (\beta-\beta')(\gamma-\gamma')\neq 0, \\ V_{13}\cup V_{23}, \ \ \ \ if \ \ (\alpha,\beta)=(\alpha',\beta') \ and \ \gamma \neq \gamma',\\ \{\lambda e_1+\mu e_2+\nu e_3; \lambda^2+\mu^2+\nu^2=1\}\ \ \ \ if \ \ (\alpha, \beta, \gamma)=(\alpha', \beta', \gamma'). \end{array}\right.\)
Corollary 4. For \(A:=\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\), we have
Case 1: \((\alpha-\alpha')(\beta-\beta')(\gamma-\gamma')\neq
0\)
Aut(A) is isomorphic to
\(\alpha=\beta=\gamma\) nd
\(\alpha'=\beta'=\gamma'\)
\(A_4\)
Otherwise
\(\mathbb{Z}_2\times
\mathbb{Z}_2\)
Case 2: \((\alpha-\alpha')(\beta-\beta')(\gamma-\gamma')=0\), \(Aut(A)\) is isomorphic to \(\mathbb{Z}_2\times\mathbb{Z}_2\)
Case 3: \((\alpha,\beta,\gamma)=(\alpha',\beta',\gamma')\)
Aut(A)
\(\alpha\neq \beta \neq \gamma \)
\(\mathbb{Z}_2\times \mathbb{Z}_2\)
\(\alpha=\beta, \beta\neq\gamma\)
\(SO(2)\)
\(\alpha=\beta=\gamma\)
\(SO(3)\)
Proposotion 9. Let \(A\) be a division algebra of eight-dimensional, the following statements are equivalent,
\(Aut(A)\) contains two distinct reflections which commute.
\(Aut(A)\) contains a subgroup isomorphic to the Klein’s group \(\mathbb{Z}_2\times \mathbb{Z}_2\).
There are vector subspaces \(X\), \(Y,\) \(Z\)
and \(T\) of two-dimensional, for which
the multiplication of \(A\) is given
by,
X
Y
Z
T
X
X
Y
Z
T
Y
Y
X
T
Z
Z
Z
T
X
Y
T
T
Z
Y
X
Proof. \((1)\Longrightarrow (2)\) This is true according to Proposition 4.
\((2)\Longrightarrow (3)\) The group \(Aut(A)\) contains two distinct reflections \(f\) and \(g\) which commute. Then there is a basis \(\{e, e_1, e_2, e_3, e_4, e_5, e_6, e_7\}\) consisting of eigenvectors common to \(f\) and \(g\) where \(E_1(f)=Lin\{e, e_1, e_2, e_3\}\) and \(E_{-1}(f)=Lin\{e_4, e_5, e_6, e_7\}\). As \(f\neq g\) the subalgebras \(E_1(f)\), \(E_1(g)\) of dimension \(4\), cannot coincide. Consequently the subalgebra \(X:=E_1(f)\cap E_1(g)\), is of dimension \(\leq 2.\)
Moreover the subalgebra \(X\) cannot be reduced to \(\mathbb{R}e.\) Otherwise the vector subspace \(Lin\{e_1,e_2, e_3\}:=E\) of \(E_1(f)\), would be contained in \(E_{-1}(g)\) and we would have \(E^2\subset E_1(f)^2 \cap E_{-1}(g)^2= E_1(f)\cap E_1(g)=\mathbb{R}e\) nonsense because, \(e_1\), \(e_2\) \(e_3\in E\) then \(e_1e_2, e_1e_3\in E^2\subset \mathbb{R}e\) so there exist \(\alpha, \beta \in \mathbb{R}\) nonzero, such that \(e_1.e_2=\alpha e\) and \(e_1e_3=\beta e\). We have \(e_1.(\beta e_2-\alpha e_3)=0\Longrightarrow \beta e_2-\alpha e_3=0\), as \(A\) is of division so \(\beta e_2=\alpha e_3\) contradicting the fact that \(e_2\) and \(e_3\) are linearly independent. So \(X\) is of dimension 2.
If for example \(X=Lin\{e, e_1\}\) we can state that \(E_1(g)=Lin\{e, e_1, e_4, e_5\}\) and \(E_{-1}(g)=Lin\{e_2, e_3, e_6, e_7\}\), we then obtain the following sub-vector spaces of dimension \(2\).
\(Y:=E_1(f)\cap E_{-1}(g)=Lin\{e_2,
e_3\}\),
\(Z:=E_1(g)\cap E_{-1}(f)=Lin\{e_4,
e_5\}\),
\(T:=E_{-1}(f)\cap E_{-1}(g)=Lin\{e_6,
e_7\}\).
It is easy to show that the multiplication of \(A\) is done according to (3).
\((3)\Longrightarrow (1)\) The vector space \(A\) decomposes into a direct sum of the subspaces vector espaces \(X\), \(Y\), \(Z\), \(T\) and the two endomorphisms \(f, g: A:=X\oplus Y\oplus Z\oplus T\rightarrow X\oplus Y\oplus Z\oplus T\) defined by, for all \(u=x+y+z+t\in X\oplus Y\oplus Z\oplus T\) we have \(f(u)= x+y-z-t\) and \(g(u)= x-y+z-t\). They are distinct reflections, which commute. Thus the subgroup of \(Aut(A)\) generated by \(f\) and \(g\) is isomorphic to \(\mathbb{Z}_2\times \mathbb{Z}_2\). ◻
Example 1. Let \(A\) be a division algebra of eight-dimensional whose product in the base \(B=\{e, u_1, \ldots, u_7\}\) is given by,
| . | e | \(u_1\) | \(u_2\) | \(u_3\) | \(u_4\) | \(u_5\) | \(u_6\) | \(u_7\) |
| e | e | \(u_1\) | \(u_2\) | \(u_3\) | \(u_4\) | \(u_5\) | \(u_6\) | \(u_7\) |
| \(u_1\) | \(u_1\) | \(-e\) | \(\gamma u_3\) | \(-\beta’ u_2\) | \(\delta u_5\) | \(-\eta’u_4\) | \(\lambda u_7\) | \(-\mu’u_6\) |
| \(u_2\) | \(u_2\) | \(-\gamma’u_3\) | \(-e\) | \(\alpha u_1\) | \(\sigma u_6\) | \(\eta u_7\) | \(-\rho’u_4\) | \(-\xi’u_5\) |
| \(u_3\) | \(u_3\) | \(-\beta u_2\) | \(-\alpha’u_1\) | \(-e\) | \(\psi u_7\) | \(-\rho u_6\) | \(\tau u_5\) | \(-\chi’u_4\) |
| \(u_4\) | \(u_4\) | \(-\delta’u_5\) | \(\eta u_6\) | \(-\tau’u_7\) | \(-e\) | \(\beta’u_1\) | \(-\xi u_2\) | \(\chi u_3\) |
| \(u_5\) | \(u_5\) | \(-\lambda’u_7\) | \(\mu u_6\) | \(\rho’u_4\) | \(-\beta u_1\) | \(-e\) | \(\alpha’u_3\) | \(\delta’u_2\) |
| \(u_6\) | \(u_6\) | \(\psi’u_7\) | \(\lambda u_5\) | \(\chi’u_4\) | \(\rho u_3\) | \(-\alpha u_2\) | \(-e\) | \(\gamma’u_1\) |
| \(u_7\) | \(u_7\) | \(\xi’u_5\) | \(\mu’ u_6\) | \(\delta u_4\) | \(\chi’u_3\) | \(\tau’u_2\) | \(\eta’u_1\) | \(-e\) |
Let \(x=x_0e+\sum\limits_{i=1}^{7}x_iu_i\in A\), the endomorphisms \(f: A\longrightarrow A\) and \(g: A\longrightarrow A\) defined by \(f(x)=x_0e+x_1u_1-x_2u_2-x_3u_3+x_4u_4+x_5u_5-x_6u_6-x_7u_7\) and \(g(x)=x_0e+x_1u_1+x_2u_2+x_3u_3-x_4u_4-x_5u_5-x_6u_6-x_7u_7\) are automorphisms of \(A\) which commute. Thus \(Aut(A)\) contains a subgroup isomorphic to the Klein’s group \(\mathbb{Z}_2\times \mathbb{Z}_2\).