On Real Algebras Admitting Reflections which Commute

Diabang, André; Diompy, Mankagna; Ba, Alhousseynou

doi:10.61091/jcmcc119-05

Abstract

1. Introduction
2. Notations and Preliminary Results
3. Automorphisms of two-dimensional Division Algebra with Reflection
4. Automorphisms of four-dimensional Division Algebra with Reflection
5. Automorphisms of H(α,β,γ,α′,β′,γ′) Algèbra
6. A Note on Eight-dimensional Division Algebras having a Reflection

References

Journal of Combinatorial Mathematics and Combinatorial Computing

Volume 119
Pages: 45-52

Research article

On Real Algebras Admitting Reflections which Commute

^¹, ^², ^²

¹Département de Mathématiques, UFR. Sciences et Technologies (SET), Université Iba Der Thiam, Thiés (UIDT), Sénégal

²Déepartement de Mathématiques et Informatique, Faculté des Sciences et Techniques (FST), Universitée Cheikh Anta Diop, Dakar (UCAD), Sénéegal

Received: 12/09/2023
Accepted: 20/12/2023
Published: 31/03/2024

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Abstract

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 $Z_{2} \times Z_{2}$ .

Keywords: Division algebra, Albert-isotopy, Automorphisms group

1. Introduction

The classification of non-associative division algebras over a commutative field $K$ with a characteristic different from $2$ is a pastionning and topical problem, whose origins date back to the discovery of quaternions ( $H$ , Hamilton 1843) and octonions ( $O$ , 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 $R$ 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 $Z_{2} \times Z_{2}$ with $Z_{2} := \frac{Z}{2 Z}$ .

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.

2. Notations and Preliminary Results

Let $A$ be an arbitrary non-associative real algebra. We define, $I (A) = {x \in A, x^{2} = x}$ and let $x$ , $y \in A$ , $[x, y] = x y - y x$ and $(x, y, z) = (x y) z - x (y z)$ .

$A$ is said to be
- – division if the operations $L_{x} : A \to A$ , $y \mapsto x y$ and $R_{x} : A \to A$ , $y \mapsto y x$ 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 $\partial : A \to A$ 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 $A$ ( $E n d_{K} (A)$ ) 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 $D e r (A) .$
A linear map $f : A \to A$ 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 $A u t_{K} (A)$ for the usual law. $f \in A u t (A)$ is said to be a reflection of $A,$ if $f$ is involutive $(f \circ f = i d_{A})$ not identical $(f \neq i d_{A})$ . Let $f$ be an automorphism of $A$ and $λ \in R$ , we denote by $E_{λ} (f)$ the kernel of $f - λ i d_{A}$ , $i d_{A}$ being the identity operator of $A$ .
Let $f, g : A \to A$ 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 $x ⊙ y = f (x) g (y)$ .

Remark 1. A linearization of $(x, x, x) = 0$ , gives $[x^{2}, y] + [x y + y x, x] = 0$ 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, ${e_{1}}^{2} = e_{1}$ , $e_{1} e_{2} = e_{3}$ , $e_{2} e_{1} = e_{4}$ 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 $s p (f) = {- 1, 1}$ . 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 $d i m E_{1} (f) = n \geq 1,$ and $d i m (E_{1} (f) + E_{- 1} (f)) = 2 n .$

In addition, the following statements are equivalent,

(a) $f$ is a reflection of $A .$
(b) $A = E_{1} (f) \oplus E_{- 1} (f) .$

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 $α$ , $β,$ $γ,$ $α^{'},$ $β^{'}$ and $γ^{'}$ be real numbers. We define algebra $A$ having a base ${e, e_{1}, e_{2}, e_{3}}$ for which the product is given as

$\begin{matrix} (1) & \begin{array}{ccccc} ⊙ & e & e_{1} & e_{2} & e_{3} \\ e & e & e_{1} & e_{2} & e_{3} \\ e_{1} & e_{1} & - e & γ e_{3} & β^{'} e_{2} \\ e_{2} & e_{2} & γ^{'} e_{3} & - e & α e_{1} \\ e_{3} & e_{3} & β e_{2} & α^{'} e_{1} & - e \end{array} \end{matrix}$

We denote this algebra $A$ by $H (α, β, γ, α^{'}, β^{'}, γ^{'})$ .

In [3], the author gives the following results:

Theorem 1. A necessary condition that the algebra $H (α, β, γ, α^{'}, β^{'}, γ^{'})$ should be a division algebra, is that the six nonzero constants $α,$ $β,$ $γ,$ $- α^{'},$ $- β^{'},$ $- γ^{'}$ 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 $f (α, β, γ) = f (- α^{'}, - β^{'}, - γ^{'})$ where $f$ is defined by $f (x, y, z) = x + y + z - x y z .$

Theorem 2. A necessary and sufficient condition that the division algebra $H (α, β, γ, α^{'}, β^{'}, γ^{'})$ should have two sided rank 2 is that the equations $α^{'} = α$ , $β^{'} = β$ and $γ^{'} = γ$ 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,

$A u t (A)$ contains two distinct reflections which commute.
$A u t (A)$ contains a subgroup isomorphic to the Klein’s group $Z_{2} \times Z_{2}$ .

Proof. $(1) ⟹ (2)$ Let $A$ be a real algebra such that $A u t (A)$ contains two reflections which commute $f$ and $g .$ Then $h = f \circ g \in A u t (A)$ and is a different reflection of $f$ and $g$ . The subgroup of $A u t (A)$ generated by $f$ and $g$ is isomorphic to the klein’s group $Z_{2} \times Z_{2}$ .

$(2) ⟹ (1)$ Obvious because the elements of the subgroup of $A u t (A)$ which is isomorphic to $Z_{2} \times Z_{2}$ different from the identity are distinct reflections that commute.

3. Automorphisms of two-dimensional Division Algebra with Reflection

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 $E_{1} (f) = R e$ and $E_{- 1} (f) = R e_{1}$ .

Suppose that $E_{1} (g) = R e_{1}$ and $E_{- 1} (g) = R e$ , let $x = a e + b e_{1} \in A,$ we have $f (x) = a f (e) + b f (e_{1}) = a e - b e_{1} = - a g (e) - b g (e_{1}) = - g (a e + b e_{1}) = - g (x),$ thus $f = - g$ (which is not an automophism of $A$ ), absurd. So $E_{1} (g) = R e$ and $E_{- 1} (g) = R e_{1}$ , 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,

$\begin{matrix} (2) & \begin{array}{ccc} ⊙ & e & e_{1} \\ e & e & α e_{1} \\ e_{1} & β e_{1} & γ e \end{array} \end{matrix}$

with $α$ , $β$ , $γ$ are non-zero real numbers and $α β γ < 0$ . We will denote this algebra $A (α, β, γ)$ .

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 $(*)$ 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,

$A u t (A)$ contains a reflection.
$A$ is isomorphic to $A (α, β, γ)$ with $α β γ < 0$ .

Proof. $(1) \Rightarrow (2)$ The Theorem 3 gives the result.

$(2) \Rightarrow (1)$ The endomorphism $f$ of $A (α, β, γ)$ defined by $f (x_{0} e + x_{1} e_{1}) = x_{0} e - x_{1} e_{1}$ is a reflection of $A (α, β, γ)$ .

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 $A (α, β, γ)$ with $β = α$ and $γ < 0$ .
$A$ 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 $A (α, β, γ)$ with $α β γ < 0$ . Since $A$ is commutative, $e_{1} e = e e_{1} ⟹ α = β$ , therefore we get the result.

$(2) \Rightarrow (3)$ It is easy to show that $A (α, β, γ)$ , with $β = α$ and $γ < 0$ , is at third power-associative.

$(3) \Rightarrow (4)$ The Remark 1 gives the result.

$(4) \Rightarrow (1)$ $A$ is isomorphic to $A (α, β, γ)$ with $α β γ < 0$ and it is at $121$ power-associative, We have $(e_{1}, {e_{1}}^{2}, e_{1}) = 0 \Rightarrow γ^{2} (α - β) e = 0 \Rightarrow α = β .$ and $γ < 0.$ 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 $A (α, β, γ)$ and we have.

	$I (A)$
$α + β \neq 0 e t γ (α + β - 1) > 0$	${e, λ_{0} e - λ_{1} e_{1}, λ_{0} e + λ_{1} e_{1}}$
otherwise	${e}$

with $λ_{0} = \frac{1}{α + β}$ , ${λ_{1}}^{2} = \frac{α + β - 1}{γ (α + β)^{2}}$ .

Proof. Since $A$ is isomorphic to $A (α, β, γ)$ with $α β γ < 0.$ Let $x = λ_{0} e + λ_{1} e_{1} \in I (A)$ , we have $x^{2} = x \Leftrightarrow {\begin{cases} {λ_{0}}^{2} + γ {λ_{1}}^{2} = λ_{0}; \\ (α + β) λ_{0} λ_{1} = λ_{1}, \end{cases}$ we obtain $I (A)$ by resolving the system and discussing on $α + β$ and $\frac{α + β - 1}{γ}$ .

Corollary 2. Let $A$ be a two-dimensional division algebra and let $f$ be a reflection of $A$ , then A is isomorphic to $A (α, β, γ)$ and we have

	$A u t (A)$ is isomorphic to
$α + β \neq 0 e t γ (α + β - 1) > 0$	$Z_{2}$ or $S_{3}$
	$Z_{2}$

4. Automorphisms of four-dimensional Division Algebra with Reflection

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 $E_{1} (f) = R e + R e_{1},$ $E_{- 1} (f) = R e_{2} + R e_{3},$ $E_{1} (g) = R e + R e_{2},$ and $E_{- 1} (g) = R e_{1} + R e_{3} .$

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 $E_{1} (f) = R e + R e_{1}$ , $E_{- 1} (f) = R e_{2} + R e_{3}$ , 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 $E_{1} (g) = R e + R e_{2}$ and $E_{- 1} (g) = R e_{1} + R e_{2}$ . 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 $e_{1} \in E_{1} (f) \cap E_{1} (g) = R e$ . Thus by analogy, the elements ${e_{i}}^{2} \in E_{1} (f) \cap E_{1} (g) = R e$ 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,

$A u t (A)$ contains two distinct reflections which commute.
$A u t (A)$ contains a subgroup isomorphic to the Klein’s group $Z_{2} \times Z_{2}$ .
$A$ is isomorphic to the algebra $H (α, β, γ, α^{'}, β^{'}, γ^{'})$ .

Proof. $(1) ⟺ (2)$ This is true according to Proposition 1.

$(2) ⟺ (3)$ 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 $R e + R e_{i}$ , for all $i \in {1, 2, 3}$ , are isomorphic to the algebra $C$ since they are real algebras of unit division of two-dimensional. We can now set $E_{1} (f) = R e + R e_{1},$ $E_{- 1} (f) = R e_{2} + R e_{3},$ $E_{1} (g) = R e + R e_{2}$ and $E_{- 1} (g) = R e_{1} + R e_{3} .$

We have $e_{1} \in E_{1} (f) e t e_{2} \in E_{- 1} (f) \Rightarrow e_{1} e_{2}, e_{2} e_{1} \in E_{1} (f) . E_{- 1} (f) = E_{- 1} (f)$ , $e_{1} \in E_{- 1} (g)$ and $e_{2} \in E_{1} (g) \Rightarrow e_{1} e_{2}, e_{2} e_{1} \in E_{- 1} (g) . E_{1} (g) = E_{- 1} (g)$ .

As a result $e_{1} e_{2},$ and $e_{2} e_{1} \in E_{- 1} (f) \cap E_{- 1} (g) = R e_{3}$ . In the same way we have $\begin{aligned} e_{1} e_{3}, & e_{3} e_{1} \in E_{- 1} (f) \cap E_{1} (g) = R e_{2}, \\ e_{2} e_{3}, & e_{3} e_{2} \in E_{1} (f) \cap E_{- 1} (g) = R e_{1} . \end{aligned}$ This gives the multiplication table for (1).

$(3) ⟹ (1)$ $f : A ⟶ A;$ $λ_{0} e + \sum_{i = 1}^{3} λ_{i} e_{i} ⟼ λ_{0} e + λ_{1} e_{1} - \sum_{i = 2}^{3} λ_{i} e_{i}$ and $g : A ⟶ A;$ $λ_{0} e + \sum_{i = 1}^{3} λ_{i} e_{i} ⟼ λ_{0} e - λ_{1} e_{1} + λ_{2} e_{2} - λ_{3} e_{3}$ 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 $H (α, β, γ, α^{'}, β^{'}, γ^{'})$ 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 $f \circ R_{e} (x) = f (R_{e} (x)) = f (x e) = f (x) f (e) = f (x) e = R_{e} (f (x)) = R_{e} \circ f (x),$ so $f \circ R_{e} = R_{e} \circ f \Leftrightarrow f \circ {R_{e}}^{- 1} = {R_{e}}^{- 1} \circ f .$ As well as $f \circ {L_{e}}^{- 1} = {L_{e}}^{- 1} \circ f$ , $(A, ⊙)$ where $x ⊙ y = {R_{e}}^{- 1} (x) . {L_{e}}^{- 1} (y)$ for all $x$ , $y \in A$ , is isotope to $A$ , and is unitary of unit $e$ . We have, $f (x ⊙ y) = f ({R_{e}}^{- 1} (x) . {L_{e}}^{- 1} (y)) = f ({R_{e}}^{- 1} (x)) . f ({L_{e}}^{- 1} (y)) = {R_{e}}^{- 1} (f (x)) . {L_{e}}^{- 1} (f (y)) = f (x) ⊙ f (y),$ then $f \in A u t ((A, ⊙))$ . We also show by analogy that $g \in A u t ((A, ⊙)) .$ Hence $(A, ⊙)$ verifies the assumptions of Proposition 5 therefore it is isomorphic to $H (α, β, γ, α^{'}, β^{'}, γ^{'})$ .

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 $H (α, β, γ, α^{'}, β^{'}, γ^{'})$ , with $α^{'} = α$ , $β^{'} = β$ , $γ^{'} = γ$ .
$A$ is at third power-associative.
$A$ is at 121 power-associative.

Proof. $(1) ⟹ (2)$ For all $x = λ_{0} e + \sum_{i = 1}^{3} λ_{i} e_{i} \in H (α, β, γ, α^{'}, β^{'}, γ^{'})$ , with $α^{'} = α$ , $β^{'} = β$ , $γ^{'} = γ$ , we have $x^{2} = - N (x) e + 2 λ_{0} x$ with $N (x) = \sum_{i = 0}^{3} {λ_{i}}^{2}$ and it is easy to check that $(x, x, x) = 0$

$(2) ⟹ (3)$ It is clear from Remark 1

$(3) ⟹ (1)$ A simple calculation show that $α = α^{'}$ , $β = β^{'}$ et $γ = γ^{'}$ .

5. Automorphisms of $H (α, β, γ, α^{'}, β^{'}, γ^{'})$ Algèbra

We notice that ${1, 2, 3} := J_{3};$ ${\sum_{i = 1}^{3} α_{i} e_{i}, α_{i} \in R} := V;$ ${x \in V; x^{2} = - e} := V_{e}$ and ${λ e_{i} + μ e_{j}; λ^{2} + μ^{2} = 1} := V_{i j} i j \in J_{3}, i \neq j .$

Proposition 5. Let $D = {x \in H (α, β, γ, α^{'}, β^{'}, γ^{'}); x^{2} = - e}$ . If $(α - α^{'}) (β - β^{'}) (γ - γ^{'}) = 0$ , then $D = {λ_{1} e_{1} + λ_{2} e_{2} + λ_{3} e_{3}; \sum_{i = 1}^{3} {λ_{i}}^{2} = 1}$ , otherwise, $D = {- e_{1}, e_{1}, - e_{2}, e_{2}, - e_{3}, e_{3}}$

Proof. Let $x = λ_{0} e + \sum_{i = 1}^{3} λ_{i} e_{i}$ and $x^{'} = λ_{0}^{'} e + \sum_{i = 1}^{3} λ_{i}^{'} e_{i} \in$ $H (α, β, γ, α^{'}, β^{'}, γ^{'})$ , we have $x x^{'} = (λ_{0} λ_{0}^{'} - λ_{1} λ_{1}^{'} - λ_{2} λ_{2}^{'} - λ_{3} λ_{3}^{'}) e + (λ_{0} λ_{1}^{'} + λ_{0}^{'} λ_{1} + α λ_{2} λ_{3}^{'} - α^{'} λ_{3} λ_{2}^{'}) e_{1} + (λ_{0} λ_{2}^{'} + λ_{0}^{'} λ_{2} + β λ_{3} λ_{1}^{'} - β^{'} λ_{1} λ_{3}^{'}) e_{2} + (λ_{0} λ_{3}^{'} + λ_{0}^{'} λ_{3} + γ λ_{1} λ_{2}^{'} - γ^{'} λ_{2} λ_{1}^{'}) e_{3} .$ Especially $x^{2} = ({λ_{0}}^{2} - {λ_{1}}^{2} - {λ_{2}}^{2} - {λ_{3}}^{2}) e + (2 λ_{0} λ_{1} + (α - α^{'}) λ_{2} λ_{3}) e_{1} + (2 λ_{0} λ_{2} + (β - β^{'}) λ_{1} λ_{3}) e_{2} + (2 λ_{0} λ_{3} + (γ - γ^{'}) λ_{1} λ_{2}) e_{3} .$ So $x \in D ⟺ {\begin{cases} {λ_{0}}^{2} - {λ_{1}}^{2} - {λ_{2}}^{2} - {λ_{3}}^{2} = - 1; \\ 2 λ_{0} λ_{1} + (α - α^{'}) λ_{2} λ_{3} = 0; \\ 2 λ_{0} λ_{2} + (β - β^{'}) λ_{1} λ_{3} = 0; \\ 2 λ_{0} λ_{3} + (γ - γ^{'}) λ_{1} λ_{2} = 0, \end{cases}$ we obtain $D$ by resolving the system and discussing on $(α - α^{'}) (β - β^{'}) (γ - γ^{'}) = 0$ and $(α - α^{'}) (β - β^{'}) (γ - γ^{'}) \neq 0$ .

Proposition 6. The group of automorphisms of algebra $H (α, β, γ, α^{'}, β^{'}, γ^{'})$ contains two reflections which commute. Furthermore, the following two propositions are equivalent:

$α = α^{'} = β = β^{'} = γ = γ^{'}$ .
$H (α, β, γ, α^{'}, β^{'}, γ^{'})$ is isomorphic to $H^{(\frac{α + 1}{2})} .$

Proof. Proposition 4 shows that $H (α, β, γ, α^{'}, β^{'}, γ^{'})$ contains two reflections which commute.

Lemma 2. Let $f \in A u t (H (α, β, γ, α^{'}, β^{'}, γ^{'}))$ , we have

$f (e) = e$ ,
$f (e_{i})^{2} = - e$ for all $i \in J_{3}$ ,
${\begin{cases} γ^{'} f (e_{1}) f (e_{2}) + γ f (e_{2}) f (e_{1}) = 0, E (3.1) \\ β f (e_{1}) f (e_{3}) + β^{'} f (e_{3}) f (e_{1}) = 0, E (3.2) \\ α^{'} f (e_{2}) f (e_{3}) + α f (e_{3}) f (e_{2}) = 0, E (3.3) \end{cases}$

Lemma 3. Let $f \in A u t (H (α, β, γ, α^{'}, β^{'}, γ^{'}))$ , then $f (V) \subseteq V .$

Proof. The demonstration is reduced to the case $(α - α^{'}) (β - β^{'}) (γ - γ^{'}) \neq 0$ (Proposition 5, Lemma 2). We will show that ${f (e_{1}), f (e_{2}), f (e_{3})} \subseteq V$ .

Let $f (e_{i}) = λ_{0 i} e + λ_{1 i} e_{1} + λ_{2 i} e_{2} + λ_{3 i} e_{3}$ 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 $λ_{02} λ_{03} - λ_{12} λ_{13} - λ_{22} λ_{23} - λ_{23} λ_{33} := λ$ .

If $f (e_{1}) \notin V$ , since $α f (e_{1}) = f (e_{2}) f (e_{3}) \notin V$ , then $λ \neq 0$ . E(3.3) shows that $α - α^{'} = 0$ 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 $(α - α^{'}) (β - β^{'}) (γ - γ^{'}) \neq 0$ . Let $f \in A u t (H (α, β, γ, α^{'}, β^{'}, γ^{'}))$ . Then there exists a permutation $σ$ of the set $J_{3}$ , which is not a transposition, and scalars $ε_{1},$ $ε_{2} \in {1, - 1}$ such that $(f (e_{1}), f (e_{2}), f (e_{3})) = (ε_{1} e_{σ (1)}, ε_{2} e_{σ (2)}, ε_{3} e_{σ (3)})$ . Furthermore, the following two are equivalent:

$A u t (H (α, β, γ, α^{'}, β^{'}, γ^{'}))$ contains an element $f$ which transitively permutes, $\pm e_{1}$ , $\pm e_{2}$ and $\pm e_{3}$ .
$α = β = γ \neq α^{'} = β^{'} = γ^{'}$ .

Proposotion 8. We have

$V_{e} = {\begin{cases} {\pm e_{1}, \pm e_{2}, \pm e_{3}}, i f (α - α^{'}) (β - β^{'}) (γ - γ^{'}) \neq 0, \\ {\pm e_{1}} \cup V_{23}, i f α = α^{'} a n d (β - β^{'}) (γ - γ^{'}) \neq 0, \\ V_{13} \cup V_{23}, i f (α, β) = (α^{'}, β^{'}) a n d γ \neq γ^{'}, \\ {λ e_{1} + μ e_{2} + ν e_{3}; λ^{2} + μ^{2} + ν^{2} = 1} i f (α, β, γ) = (α^{'}, β^{'}, γ^{'}) . \end{cases}$

Corollary 4. For $A := H (α, β, γ, α^{'}, β^{'}, γ^{'})$ , we have

Case 1: $(α - α^{'}) (β - β^{'}) (γ - γ^{'}) \neq 0$

	Aut(A) is isomorphic to
$α = β = γ$ nd $α^{'} = β^{'} = γ^{'}$	$A_{4}$
Otherwise	$Z_{2} \times Z_{2}$

Case 2: $(α - α^{'}) (β - β^{'}) (γ - γ^{'}) = 0$ , $A u t (A)$ is isomorphic to $Z_{2} \times Z_{2}$

Case 3: $(α, β, γ) = (α^{'}, β^{'}, γ^{'})$

	Aut(A)
$α \neq β \neq γ$	$Z_{2} \times Z_{2}$
$α = β, β \neq γ$	$S O (2)$
$α = β = γ$	$S O (3)$

6. A Note on Eight-dimensional Division Algebras having a Reflection

Proposotion 9. Let $A$ be a division algebra of eight-dimensional, the following statements are equivalent,

$A u t (A)$ contains two distinct reflections which commute.
$A u t (A)$ contains a subgroup isomorphic to the Klein’s group $Z_{2} \times Z_{2}$ .

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. $(1) ⟹ (2)$ This is true according to Proposition 4.

$(2) ⟹ (3)$ The group $A u t (A)$ 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 $E_{1} (f) = L i n {e, e_{1}, e_{2}, e_{3}}$ and $E_{- 1} (f) = L i n {e_{4}, e_{5}, e_{6}, e_{7}}$ . 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 $R e .$ Otherwise the vector subspace $L i n {e_{1}, e_{2}, e_{3}} := E$ of $E_{1} (f)$ , would be contained in $E_{- 1} (g)$ and we would have $E^{2} \subset E_{1} (f)^{2} \cap E_{- 1} (g)^{2} = E_{1} (f) \cap E_{1} (g) = R e$ nonsense because, $e_{1}$ , $e_{2}$ $e_{3} \in E$ then $e_{1} e_{2}, e_{1} e_{3} \in E^{2} \subset R e$ so there exist $α, β \in R$ nonzero, such that $e_{1} . e_{2} = α e$ and $e_{1} e_{3} = β e$ . We have $e_{1} . (β e_{2} - α e_{3}) = 0 ⟹ β e_{2} - α e_{3} = 0$ , as $A$ is of division so $β e_{2} = α e_{3}$ contradicting the fact that $e_{2}$ and $e_{3}$ are linearly independent. So $X$ is of dimension 2.

If for example $X = L i n {e, e_{1}}$ we can state that $E_{1} (g) = L i n {e, e_{1}, e_{4}, e_{5}}$ and $E_{- 1} (g) = L i n {e_{2}, e_{3}, e_{6}, e_{7}}$ , we then obtain the following sub-vector spaces of dimension $2$ .

$Y := E_{1} (f) \cap E_{- 1} (g) = L i n {e_{2}, e_{3}}$ ,
$Z := E_{1} (g) \cap E_{- 1} (f) = L i n {e_{4}, e_{5}}$ ,
$T := E_{- 1} (f) \cap E_{- 1} (g) = L i n {e_{6}, e_{7}}$ .

It is easy to show that the multiplication of $A$ is done according to (3).

$(3) ⟹ (1)$ The vector space $A$ decomposes into a direct sum of the subspaces vector espaces $X$ , $Y$ , $Z$ , $T$ and the two endomorphisms $f, g : A := X \oplus Y \oplus Z \oplus T \to X \oplus Y \oplus Z \oplus T$ 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 $A u t (A)$ generated by $f$ and $g$ is isomorphic to $Z_{2} \times Z_{2}$ .

Example 1. Let $A$ be a division algebra of eight-dimensional whose product in the base $B = {e, u_{1}, \dots, u_{7}}$ is given by,

.	e	$u_{1}$	$u_{2}$	$u_{3}$	$u_{4}$	$u_{5}$	$u_{6}$	$u_{7}$
e	e	$u_{1}$	$u_{2}$	$u_{3}$	$u_{4}$	$u_{5}$	$u_{6}$	$u_{7}$
$u_{1}$	$u_{1}$	$- e$	$γ u_{3}$	$- β^{'} u_{2}$	$δ u_{5}$	$- η^{'} u_{4}$	$λ u_{7}$	$- μ^{'} u_{6}$
$u_{2}$	$u_{2}$	$- γ^{'} u_{3}$	$- e$	$α u_{1}$	$σ u_{6}$	$η u_{7}$	$- ρ^{'} u_{4}$	$- ξ^{'} u_{5}$
$u_{3}$	$u_{3}$	$- β u_{2}$	$- α^{'} u_{1}$	$- e$	$ψ u_{7}$	$- ρ u_{6}$	$τ u_{5}$	$- χ^{'} u_{4}$
$u_{4}$	$u_{4}$	$- δ^{'} u_{5}$	$η u_{6}$	$- τ^{'} u_{7}$	$- e$	$β^{'} u_{1}$	$- ξ u_{2}$	$χ u_{3}$
$u_{5}$	$u_{5}$	$- λ^{'} u_{7}$	$μ u_{6}$	$ρ^{'} u_{4}$	$- β u_{1}$	$- e$	$α^{'} u_{3}$	$δ^{'} u_{2}$
$u_{6}$	$u_{6}$	$ψ^{'} u_{7}$	$λ u_{5}$	$χ^{'} u_{4}$	$ρ u_{3}$	$- α u_{2}$	$- e$	$γ^{'} u_{1}$
$u_{7}$	$u_{7}$	$ξ^{'} u_{5}$	$μ^{'} u_{6}$	$δ u_{4}$	$χ^{'} u_{3}$	$τ^{'} u_{2}$	$η^{'} u_{1}$	$- e$

Let $x = x_{0} e + \sum_{i = 1}^{7} x_{i} u_{i} \in A$ , the endomorphisms $f : A ⟶ A$ and $g : A ⟶ A$ defined by $f (x) = x_{0} e + x_{1} u_{1} - x_{2} u_{2} - x_{3} u_{3} + x_{4} u_{4} + x_{5} u_{5} - x_{6} u_{6} - x_{7} u_{7}$ and $g (x) = x_{0} e + x_{1} u_{1} + x_{2} u_{2} + x_{3} u_{3} - x_{4} u_{4} - x_{5} u_{5} - x_{6} u_{6} - x_{7} u_{7}$ are automorphisms of $A$ which commute. Thus $A u t (A)$ contains a subgroup isomorphic to the Klein’s group $Z_{2} \times Z_{2}$ .

References:

Bott, R., and Milnor, J., 1958. On the parallelizability of the spheres. Bulletin of the American Mathematical Society 64, pp.87-89.
Diabang, A.S., Diankha, O. and Rochdi, A., 2016. On the automorphisms of absolute-valued algebras. International Journal of Algebra, 10(3), pp.113-123.
Bruck, R.H., 1944. Some results in the theory of linear non-associative algebras. Transactions of the American Mathematical Society, 56, pp.141-199.
Althoen, S.C. and Kugler, L.D., 1983. When is $R^{2}$ a division algebra?. The American Mathematical Monthly, 90(9), pp.625-635.

[1] Bott, R., and Milnor, J., 1958. On the parallelizability of the spheres. Bulletin of the American Mathematical Society 64, pp.87-89.

[2] Diabang, A.S., Diankha, O. and Rochdi, A., 2016. On the automorphisms of absolute-valued algebras. International Journal of Algebra, 10(3), pp.113-123.

[3] Bruck, R.H., 1944. Some results in the theory of linear non-associative algebras. Transactions of the American Mathematical Society, 56, pp.141-199.

[4] Althoen, S.C. and Kugler, L.D., 1983. When is $R^{2}$ a division algebra?. The American Mathematical Monthly, 90(9), pp.625-635.

Contents

Journal of Combinatorial Mathematics and Combinatorial Computing

On Real Algebras Admitting Reflections which Commute

Abstract

1. Introduction

2. Notations and Preliminary Results

3. Automorphisms of two-dimensional Division Algebra with Reflection

4. Automorphisms of four-dimensional Division Algebra with Reflection

5. Automorphisms of $H (α, β, γ, α^{'}, β^{'}, γ^{'})$ Algèbra

6. A Note on Eight-dimensional Division Algebras having a Reflection

References:

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Guidelines

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Contents

Journal of Combinatorial Mathematics and Combinatorial Computing

On Real Algebras Admitting Reflections which Commute

Abstract

1. Introduction

2. Notations and Preliminary Results

3. Automorphisms of two-dimensional Division Algebra with Reflection

4. Automorphisms of four-dimensional Division Algebra with Reflection

5. Automorphisms of H(α,β,γ,α′,β′,γ′) Algèbra

6. A Note on Eight-dimensional Division Algebras having a Reflection

References:

Information

Guidelines

CP Initiatives

Follow CP

5. Automorphisms of $H (α, β, γ, α^{'}, β^{'}, γ^{'})$ Algèbra