Klassifikationsproblemet för reella flexibla divisionsalgebror

On the Classification of the Real Flexible Division Algebras

Erik Darp¨ o

Matematiska Institutionen, Uppsala Universitet, Box 480, S-75106 Uppsala, Sweden.

erik.darpo@math.uu.se

Abstract

We classify the real commutative division algebras, completing an app- roach by Althoen and Kugler. We solve the isomorphism problem for scalar isotopes of real quadratic division algebras, and classify the generalized pseudo- octonion algebras. In view of earlier results by Benkart, Britten and Osborn and Cuenca Mira et al., this reduces the problem of classifying the real flexi- ble division algebras to the normal form problem for the natural action of the group G

on the set of positive definite symmetric linear endomorphisms of R

.

In addition, the automorphism groups of the real flexible division algebras are described.

Mathematics Subject Classification 2000: 15A21, 17A20, 17A35, 17A36, 17A45.

Keywords: Real division algebra, flexible algebra, scalar isotope, generalized pseudo- octonion, automorphism, group action.

1 Introduction

Let k be a field. A k-algebra is understood to be a vector space A over k, endowed with a bilinear multiplication mapping A × A → A, (x, y) 7→ xy. The algebra A is said to be a division algebra if A 6= {0} and the linear endomorphisms L a : A → A, x 7→ ax and R _a : A → A, x 7→ xa are bijective for all a ∈ A r {0}. In case A is finite-dimensional, this is equivalent to saying that A has no zero divisors, i.e.

xy = 0 only if x = 0 or y = 0. Finite-dimensional real division algbras are known to have dimension either 1, 2, 4 or 8 (Bott and Milnor [4], Kervaire [10]). All algebras considered in this paper are assumed to be finite-dimensional.

Denote by A ^± the category of triples (X, •, [, ]), where X is a vector space over k, and • and [, ] are commutative and anti-commutative algebra structures on X respectively. Morphisms in A ^± are those linear mappings that respect both struc- tures.

Given any k-algebra A, define [x, y] = xy − yx and x • y = xy + yx. The assign- ment A 7→ A ^± = (A, •, [, ]) defines a functor ? ^± from the category of k-algebras to A ^± , acting on morphisms identically. If chark 6= 2, then ? ^± is an isomorphism of categories, with inverse (X, •, [, ]) 7→ (X, µ); µ(x, y) = ¹ ₂ (x • y + [x, y]).

Let K be a category. For X ∈ K, [X] denotes the isomorphism class of X. A cross-

section for K/ ∼ = is a set C ⊂ K with the property that for all X ∈ K there exists a

unique C ∈ C such that X ∼ = C. A classification of K is an explicit description of a

cross-section for K/ ∼ =.

Given any set X, I denotes the identity mapping on X. We write I ⁿ for the identity matrix of size n × n. For any Euclidean vector space V , we denote by Pds(V ) the set of positive definite symmetric endomorphisms of V .

An algebra is called flexible if x(yx) = (xy)x for any two elements x and y. A classification of the real flexible division algebras will in a natural way generalize famous theorems by Frobenius [8] and Zorn [15] stating that the associative and al- ternative ¹ real division algebras are classified by the sets {R, C, H} and {R, C, H, O}

respectively. In our Theorem 1.4 an exhaustive family of real flexible division alge- bras is presented, and the irredundancy problem is solved up to the normal form problem for the natural right group action of G ₂ = Aut(π) on P ds(R ⁷ ), where π is an arbitrary vector product on R ⁷ .

Another class of algebras, that generalizes the alternative ones, is formed by the quadratic algebras. Recall that an algebra A is quadratic if it has an identity element 1 6= 0, and if the set {1, x, x ² } is linearly dependent for all x ∈ A. It is known that a real division algebra is quadratic if and only if it is power associative.

(Dieterich [5]).

In any quadratic algebra B over R, the subset

ImB = {b ∈ B r R1 | b ² ∈ R1} ∪ {0} ⊂ B

of purely imaginary elements is a linear subspace of B, and B = R1 ⊕ ImB (Frobe- nius [11]). We shall write α + v instead of α1 + v when referring to elements in this decomposition.

Let V = (V, h i) be a finite-dimensional Euclidean space. A linear map η : V ∧V → V is called a flexible dissident map on V if, for all non-proportional v, w ∈ V , η(v∧w) is orthogonal to v and w. The class of flexible dissident maps is given the structure of a category, denoted D ^{f l} , by declaring as morphisms (V, η) → (V ⁰ , η ⁰ ) those isometric ² linear mappings σ : V → V ⁰ that satisfy ση = η ⁰ (σ ∧ σ).

Every flexible dissident map η on V determines a quadratic algebra R(η) = R × V with multiplication (α, v)(β, w) = (αβ − hv, wi, αw + βv + η(v ∧ w)). Defin- ing (Rσ)(α, v) = (α, σ(v)), we establish R as a functor from D ^{f l} to the category of quadratic algebras ³ . Denoting the category of real flexible quadratic division algebras by Q ^{f l} , Osborns theorem ([13], cf. also [6]) yields the following result.

Proposition 1.1 The functor R induces an equivalence of categories R : D ^{f l} → Q ^{f l} .

Let B be a real quadratic algebra, and λ a nonzero real number. Then the scalar isotope of B determined by λ, denoted λ B = (B, ?) is defined by

(α + v) ? (β + w) = (α + λv)(β + λw) , α, β ∈ R, v, w ∈ ImB.

A straightforward calculation shows that λ B is flexible if B is flexible, and it is obviously a division algebra whenever B is.

Next, recall that su _n C denotes the simple real Lie algebra of n × n complex anti- hermitean matrices of trace zero, under the anti-commutative structure [, ]. For each δ ∈ R r {0}, the vector space su 3 C with the multiplication

x ∗ y = δ[x, y] + i 2



xy + yx − 2

3 tr(xy)I 3



1

An algebra is alternative if any subalgebra generated by two elements is associative.

2

That is, satisfying σ(hx, yi) = hσ(x), σ(y)i for all x and y.

3

Morphisms of quadratic algebras are assumed to preserve the identity element.

is a real flexible division algebra [3] of dimension 8, which we denote by O δ . Benkart and Osborn call these algebras generalized pseudo-octonions, or GP-algebras.

We now recall the main theorem in [2] by Benkart, Britten and Osborn.

Theorem 1.2 [2, theorem 1,4] A real algebra A is a flexible division algebra if and only if it is either

1. a commutative division algebra of dimension on or two, or

2. a scalar isotope _λ B of some real flexible quadratic division algebra B, or 3. isomorphic to a generalized pseudo-octonion algebra.

As λ B is commutative if and only if B is commutative, and as C is the only quadratic division algebra of dimension two, it follows that the real commutative division algebras are precisely the flexible ones of dimension one or two. Any scalar isotope of a quadratic algebra has a central idempotent, which distinguishes these from the GP-algebras.

It is easily shown, using Proposition 1.1, that the set {R(λπ 3 )} λ>0 , i.e. R × R ³ with the multiplication

(α, v)(β, w) = (αβ − hv, wi, αv + βv + λπ 3 (v ∧ w))

classifies the real flexible quadratic division algebras of dimension 4. Here π ₃ : R ³ ∧ R ³ → R ³ denotes the standard vector product.

In [12], Cuenca Mira et al. introduce vectorial isotopy, a method by which all real flexible quadratic division algebras are constructed. Using the language of dissident maps, their main result can be formulated as follows (? ^∗ denotes the adjoint operator).

Proposition 1.3 Let π : R ⁷ ∧ R ⁷ → R ⁷ be a vector product (i.e. R(π) ∼ = O), and η any flexible dissident map on a real vector space V of dimension 7. Then

1. For any  ∈ GL(V ),  ^∗ η( ∧ ) is a flexible dissident map.

2. η ∼ = δπ(δ ∧ δ) for some δ ∈ P ds(R ⁷ ).

3. For δ 1 , δ 2 ∈ P ds(R ⁷ ), δ 1 π(δ 1 ∧ δ 1 ) ∼ = δ 2 π(δ 2 ∧ δ 2 ) if and only if δ 1 = σ ⁻¹ δ 2 σ for some σ ∈ Aut(π).

Hence classifying the seven-dimensional flexible dissident maps, and thereby the real flexible quadratic division algebras of dimension 8, completely and irredundantly, is equivalent to solving the normal form problem for the right group action

P ds(R ⁷ ) × Aut(π) → P ds(R ⁷ ), (δ, σ) → δ · σ = σ ⁻¹ δσ where π is a fixed vector product on R ⁷ .

The principal theorem of the present article, Theorem 1.4, classifies the real flex- ible division algebras up to the above normal form problem. The solution of this problem, which involves considerable technical difficulties, is postponed to a later publication.

Let A E (a, b) and A F (a, b) be the algebras defined on R ² by the multiplication tables

1 and 2 respectively.

· e ₁ e ₂ e 1 e 1 ae 1 + be 2

e 2 ae 1 + be 2 −e 1

Table 1: A E (a, b)

· e ₁ e ₂

e 1 e 1 ae 1 + be 2

e 2 ae 1 + be 2 e 2

Table 2: A F (a, b)

Theorem 1.4 Let N ∈ P ds(R ⁷ ) be a cross-section for the orbit set P ds(R ⁷ )/Aut(π).

Then the set

{R} ˙∪

{A _E (a, b) | a > 0; b > a + 1 2 } ˙ ∪ {A F (a, b) | a > 1

2 ; b > a; (a, b) 6= ( 1 2 , 1

2 )} ˙ ∪ {A F (a, b) | a, b < 0; a 6 b 6 1

2a − 1 2 } ˙ ∪ { λ R(µπ 3 ) | (λ, µ) ∈ (R r {0}) × R ^>0 } ˙ ∪ { _λ R(δπ ₇ (δ ∧ δ)) | (λ, δ) ∈ (R r {0}) × N } ˙∪

{O δ } δ>0

classifies the real flexible division algebras.

The proof of Theorem 1.4 is given in sections 2–4. The real commutative division algebras are classified in section 2. In section 3, we show that _λ A ∼ = µ B if and only if λ = µ and A ∼ = B for quadratic division algebras A and B of dimension greater than 2. In section 4, we solve the irredundancy problem for the generalized pseudo-octonions by showing that O _γ ∼ = O δ if and only if γ = ±δ.

As an additional information, in section 5 we give the automorphism groups of the real flexible division algebras.

An alternative approach to the classification of the real commutative division alge- bras is given in a forthcoming article by Darp¨ o and Dieterich ⁴ .

2 Commutative division algebras

In [1], Althoen and Kugler present some results aiming at a classification of all real two-dimensional division algebras. Here, we will complete their task for the commutative case.

We shall use the following result by Segre.

Proposition 2.1 [14, Theorem 1] Every finite-dimensional real or complex algebra with the property that x ² = 0 implies x = 0 has at least one idempotent ⁵

4

Darp¨ o and Dieterich: Classification of the real commutative division algebras. U.U.D.M.

Report 2003:34, Uppsala University, Sweden. 2003.

5

Idempotents are understood to be non-zero.

The same result is proved in [1] for real division algebras of dimension 2.

Accordingly, if A is commutative, it has a basis (u, v) such that the multiplication is given by table 3.

· u v

u u au + bv

v au + bv cu + dv a, b, c, d ∈ R.

Table 3: Commutative division algebras.

Proposition 2.2 [1, Theorem 3] A real algebra given by table 3 is a division algebra if and only if d ² < 4b

a b c d

    .

First, we consider the case when A has exactly one idempotent.

Lemma 2.3 Let A be a real commutative division algebra, and u ∈ A an idempo- tent. Then there exists an element w 6∈ [u] such that w ² = ±u.

Proof: Let (u, v) be a basis for A, such that the multiplication is given by table 3. If d = 0, the statement follows directly. Suppose d 6= 0. Then, for an arbitrary element w = αu + βv , β 6= 0 in A r [u] we have

w ² = (αu + βv) ² = α ² u ² + 2αβuv + β ² v ² = (α ² + 2αβa + β ² c)u + (2αβb + β ² d)v Hence w ² = ±u if and only if

 α ² + 2αβa + β ² c = ±1 2αβb + β ² d = 0 Using Proposition 2.2, we see that





 α =

r 1

4ab d

−

d2

−1

β = − ^2αb _d is a solution of the system, for which β 6= 0.

 The following result is a direct consequence of [1, (7) p. 629].

Lemma 2.4 An algebra determined by table 3 has exactly one idempotent if and only if (2a − d) ² < 4c(1 − 2b).

Taking into account Proposition 2.2, Lemma 2.3 and Lemma 2.4, we conclude that the basis (u, v) can be choosen in such a way that c = −1, d = 0 and a ² + 1 < 2b.

Moreover, if an algebra has such a basis, it is a commutative division algebra with exactly one idempotent.

Recall that A E (a, b) denotes R ² with multiplication given by table 1.

Proposition 2.5 1. The set E = {A _E (a, b) | a > 0, b > ^a

₂ ⁺¹ } classifies the two-

dimensional real commutative division algebras having exactly one idempotent.

2. The automorphism group of A E (a, b) ∈ E equals {I, (e ¹ , e 2 ) 7→ (e 1 , −e 2 )} if a = 0 and {I} otherwise.

Proof: Let A = A E (a, b), B = A E (a ⁰ , b ⁰ ) and let ϕ : A → B be an isomorphism.

Then ϕ(e 1 ) = e 1 . We have

(ϕ(e 2 )) ² = ϕ(e ² ₂ ) = −ϕ(e 1 ) = −e 1 = e ² ₂ 0 = (ϕ(e 2 )) ² − e ² ₂ = (ϕ(e 2 ) − e 2 )(ϕ(e 2 ) + e 2 ) and hence ϕ(e 2 ) = ±e 2 .

If ϕ(e 2 ) = e 2 : Then ϕ = I, and (a ⁰ , b ⁰ ) = (a, b).

If ϕ(e 2 ) = −e 2 : Because

ϕ(e ₁ e ₂ ) = ϕ(ae ₁ + be ₂ ) = ae ₁ − be ₂ ϕ(e ₁ )ϕ(e ₂ ) = −e ₁ e ₂ = −a ⁰ e ₁ − b ⁰ e ₂

it follows that (a ⁰ , b ⁰ ) = (−a, b). Conversely, it is clear that A _E (a, b) → A _E (−a, b), (e ₁ , e ₂ ) 7→ (e ₁ , −e ₂ ) is an isomorphism. This proves the first statement of the proposition.

From the above it is also clear that (e 1 , e 2 ) 7→ (e 1 , −e 2 ) is an automorphism of A E (a, b) if and only if a = 0, which gives the second statement.  Now, we consider the several idempotent-case. Choosing a basis of idempotents, and using Proposition 2.2, it is clear that the set K = {(a, b) ∈ R ² | ab > ¹ ₄ } parametrizes an exhaustive family {A F (a, b)} _(a,b)∈K of division algebras of the ap- propriate type.

Lemma 2.6 The algebra A F (a, b), (a, b) ∈ K has precisely two idempotents if a = ¹ ₂ or b = ¹ ₂ , and precisely three idempotents otherwise.

Proof: The elements e ₁ and e ₂ are certainly idempotents. Let w = αe ₁ + βe ₂ be an arbitrary element in A _F (a, b). We want to find solutions of the equation w ² = w for which α, β 6= 0. Under this assumption,

w ² = w ⇔  1 2a 2b 1

 α β



= 1 1



As (a, b) ∈ K and thereby 1 − 4ab 6= 0, the equation has the unique solution (α, β) =  2a − 1

4ab − 1 , 2b − 1 4ab − 1



where α, β 6= 0 precisely when a, b 6= ¹ ₂ . 

Define K n = {(a, b) ∈ K | A F (a, b) has n idempotents}.

Any isomorphism of an algebra A permutes its idempotents. If A has idempotents e 1 , . . . , e n , dim A 6 n, every permutation σ ∈ S ⁿ defines a linear automorphism ϕ σ

on A by e j 7→ e _σ(j) for j 6 dim A. In our case, as n 6 dim A + 1, the mapping σ 7→ ϕ σ is injective. This gives rise to a group action K n × S n → K n defined by

(a, b) · σ = (a ⁰ , b ⁰ ) ⇔ ϕ σ : A F (a, b) → A F (a ⁰ , b ⁰ ) is an algebra isomorphism.

x, y ∈ K _n parametrize isomorphic algebras if and only if they lie in the same orbit

under the S _n -action. Hence, classifying the real commutative division algebras

having more than one idempotent is equivalent to finding normal forms for K 2 and K 3 under these group actions.

Automorphisms of the algebra A F (x), x ∈ K n are those ϕ σ for which σ stabilizes x.

It is easily seen that the orbits in K 2 = {(a, b) ∈ K | a = ¹ ₂ ∨ b = ¹ ₂ } are the pairs of the form {(a, b), (b, a)}.

S ₃ is generated by the the permutations (1, 2)(3) and (1)(2, 3). The first one maps each (a, b) ∈ K ₃ = K r K 2 to (b, a).

The latter permutation yields the mapping (a, b) 7→ (c, b), where c = ^a+b−1 _4ab−1 . Con- sidering the function f (x) = f (x, y) = ^x+y−1 _4xy−1 , we have that (a, b) 7→ (f (a, b), b).

f ⁰ (x) = − _(4xy−1) ^(2y−1)

, so f is decreasing for y 6= ¹ ₂ . Moreover, f (x) = x ⇔ x =

1±(2y−1)

4y , that is x = ¹ ₂ or x = ^1−y _2y . This shows that the subset {(a, b) ∈ K 3 | a, b > 1

2 or a < 1 − b 2b , b < 1

2 } of K 3 is mapped to

{(a, b) ∈ K ₃ | b > 1

2 > a or a > 1 − b 2b , b < 1

2 }

and that {(a, b) ∈ K 3 | a > ^1−b _2b , b < ¹ ₂ } is fixed by the permutation (1)(2, 3).

Using that every other element in S 3 can be written as a product of these two permutations, it is straightforward to prove the following propositions.

Proposition 2.7 1. The set {A F ( ¹ ₂ , b)} _b>

2

classifies the real commutative di- vision algebras having two idempotents.

2. The set {A _F (a, b) | a > ¹ ₂ ; b > a} ∪ {A F (a, b) | a, b < 0; a 6 b 6 _2a ¹ − ¹ ₂ } classifies the real commutative division algebras having three idempotents.

Proposition 2.8 1. If (a, b) ∈ K ₂ , Aut(A _F (a, b)) = {I}.

2. If (a, b) ∈ K 3 , Aut(A F (a, b) = {ϕ σ } σ∈S where

S =



 

 



 

 



{I, (1, 2)(3)} if a = b 6= −1 {I, (1)(2, 3)} if a = ^1−b _2b {I, (1, 3)(2)} if b = ^1−a _2a

S 3 if a = b = −1

{I} otherwise

⊂ S 3

3 Scalar isotopy

Proposition 3.1 Let A and B be quadratic division algebras of dimension greater than two, and let λ, µ 6= 0. Then λ A ∼ = µ B if and only if λ = µ and A ∼ = B. The set of isomorphisms λ A ˜ → λ B equals the set of isomorphisms A ˜ →B.

Proof: Let ϕ : λ A → µ B be an isomorphism. The identity element 1 A in A is a central idempotent in λ A. Hence ϕ(1 A ) must be a central idempotent in µ B. For x = (α + v) ∈ R1 ⊕ ImB, x ∈ Z( ^µ B) if and only if in the algebra B, vw = wv for all w ∈ ImB. From Proposition 1.1, it follows that vw = wv only if v and w are linearly dependent. As dim(ImB) > 1 we get v = 0, and therefore Z( _µ B) = [1 _B ].

Hence, ϕ(1 _A ) = 1 _B , and ϕ(ImA) = ImB.

Now, let v, w ∈ ImA be linearly independent. Then λϕ(v) = ϕ(1 A ? v) = ϕ(1 A ) ? ϕ(v) = µϕ(v); hence λ = µ. Moreover, µ ² ϕ(v)ϕ(w) = ϕ(v) ? ϕ(w) = ϕ(v ? w) = λ ² ϕ(vw) = µ ² ϕ(vw), which proves that ϕ is an isomorphism from A to B.

The converse is proved analogously, using that ϕ(1 A ) = 1 B for any morphism ϕ of

the quadratic algebras A and B. 

Corollary 3.2 If A is a quadratic division algebra of dimension 4 or 8, and λ 6= 0, then Aut( _λ A) = Aut(A).

4 Generalized pseudo-octonions

Recall the definition of the category A ^± , and the functor ? ^± . We write O ^± _δ = (O _δ , • _∗ , [, ] _∗ ), while • and [, ] refer to the ordinary matrix multiplication.

The multiplication in the commutative algebra O ⁺ = (O _δ , • _∗ ) is independent of δ and is given by x • _∗ y = i x • y − ² ₃ tr(xy)I
. The multiplication in O ⁻ _δ = (O _δ , [, ] _∗ ) is given by [x, y] _∗ = 2δ[x, y].

We now review some of the notation used by Djokovi´ c in [7]. Let g be a complex, semisimple Lie algebra. g _R denotes g considered as a real algebra. A conjugation of g is an element σ ∈ Aut(g _R ) such that σ ² = 1 and σ(iX) = −iσ(X), ∀X ∈ g. Σ denotes the set of conjugations of g. Σ is always non-empty.

A real form of g is a subalgebra g 0 ⊂ g _R such that g ∼ = g 0 ⊗ _R C. For any σ ∈ Σ, g ^σ = ker(I − σ) is a real form of g, and σ 7→ g ^σ is a bijection between Σ and the set of real forms of g.

Proposition 4.1 [7, Proposition 2.1] For σ ∈ Σ, Aut(g ^σ ) = C Aut(g) (σ). ⁶

su _n C is a real form of sl n C, the complex Lie algebra of n × n-matrices of trace zero.

We will be interested in the automorphism group of su ₃ C which, by Proposition 4.1, is a subgroup of Aut(sl ₃ C). For n > 2 we have, according to [9, p. 283], that

Aut(sl _n C) = {κ A , −κ _A ◦ (? ^T ) | A ∈ SL _n (C)}

where κ A (X) = A ⁻¹ XA, and ? ^T denotes transposition.

Corollary 4.2 Aut(su n C) = {κ A , −κ A ◦ (? ^T ) | A ∈ SU n } , n > 2

Proof: su n C = (sl n C) ^σ , where σ(X) = −X ^∗ . The statement follows, using Propo-

sition 4.1. 

Proposition 4.3 Let δ, γ ∈ R r {0}. Then O ^δ ∼ = O γ ⇔ δ = ±γ.

Proof: Let ϕ : O _δ ⁻ → O _γ ⁻ be an isomorphism. Using that the functor ? ^± is an isomorphism of categories, it suffices to show that ϕ ∈ Aut(O ⁺ ) ⇔ δ = ±γ.

The homothety h _2δ : O ⁻ _δ → su 3 C, x 7→ 2δx is an isomorphism of algebras. It follows that ψ 7→ ϕ = h ⁻¹ _2γ ψh _2δ = _γ ^δ ψ is a bijection from Aut(su ₃ C) to the set of isomorphisms O _δ ⁻ → 0 ⁻ _γ . For ψ ∈ Aut(su 3 C), we have either ψ = κ A or ψ =

−κ ◦ (? ^T ).

If ψ = κ A : Then ϕ = _γ ^δ κ A . We must ensure that ϕ respects the multiplicative

6

C denotes the centralizer; C

(x) = {g ∈ G | gx = xg}

structure of O ⁺ , that is that ϕ(x) • _∗ ϕ(y) = ϕ(x • _∗ y). We have ϕ(x) • _∗ ϕ(y) = δ

γ κ _A (x) • _∗ δ

γ κ _A (y) = δ ²

γ ² i(κ _A (x) • κ _A (y) − 2

3 tr(κ _A (x)κ _A (y))I) =

= δ ²

γ ² i(κ A (x•y) − 2

3 tr(xy)κ A (I)) = δ ²

γ ² κ A (x • ∗ y) ϕ(x • _∗ y) = δ

γ κ _A (x • _∗ y)

This means that ϕ is an automorphism of O ⁺ if and only if ^δ _γ

κ _A (x• _∗ y) = _γ ^δ κ _A (x• _∗ y) for all x, y ∈ O ⁺ ; that is if and only if δ = γ.

If ψ = −κ A ◦ (? ^T ): Then ϕ = − ^δ _γ κ A ◦ (? ^T ), and

ϕ(x • _∗ y) = − δ

γ κ A (i(x•y − 2

3 tr(xy)I)) ^T = − δ

γ κ A (i(x ^T •y ^T − 2

3 tr(xy)I)) =

= − δ

γ κ _A (i(x ^T •y ^T − 2

3 tr(x ^T y ^T )I)) = − δ

γ κ _A (x ^T • ∗ y ^T ) ϕ(x) • _∗ ϕ(y) = − δ

γ κ A (x ^T ) • _∗ (− δ

γ )κ A (y ^T ) = δ ²

γ ² κ A (x ^T ) • _∗ κ A (y ^T ) =

= δ ²

γ ² κ _A (x ^T • _∗ y ^T )

Here, ϕ ∈ Aut(O ⁺ ) precisely when δ = −γ. 

As a consequence of the proof, we get

Corollary 4.4 1. Aut(O _δ ) = {κ _A | A ∈ SU 3 }

2. The set of isomorphisms O _δ →O ˜ −δ equals {κ _A ◦ (? ^T ) | A ∈ SU ₃ }

5 Automorphisms of flexible division algebras

In order to describe the automorphism groups of all real flexible division algebras, it remains to consider the quadratic ones.

First, let η = λπ ₃ , λ 6= 0 and σ ∈ O ₃ . Then σ ∈ Aut(η) ⇔ ση = η(σ ∧σ) ⇔ λσπ ₃ = λπ ₃ (σ∧σ) ⇔ σ ∈ Aut(π ₃ ). Hence, by Proposition 1.1, Aut(R(η)) = Aut(H) = SO 3 . Second, consider η = δπ(δ ∧ δ), where δ ∈ P ds(R ⁷ ) and π is a vector product on R ⁷ . Let σ ∈ O 7 . As δ = δ ^∗ , we get

σ ∈ Aut(η) ⇔ ση = η(σ ∧ σ) ⇔

⇔ σδπ(δ ∧ δ) = δπ((δσ) ∧ (δσ)) ⇔

⇔ δπ(δ ∧ δ) = (δσ) ^∗ π((δσ) ∧ (δσ)) ⇔

⇔ π = (δσδ ⁻¹ ) ^∗ π((δσδ ⁻¹ ) ∧ (δσδ ⁻¹ )) ⇔ δσδ ⁻¹ ∈ Aut(π)

Hence we have that Aut(η) = δ ⁻¹ Aut(π)δ ∩ O 7 .

Lemma 5.1 With the above notation; δ ⁻¹ Aut(π)δ ∩ O 7 = C _Aut(π) (δ).

Proof: Let v = {v 1 , . . . , v 7 } be an ON-eigenbasis for R ⁷ with respect to δ, with

corresponding eigenvalues λ 1 > . . . > λ ⁷ . Let f ∈ Aut(π) be such that δ ⁻¹ f δ ∈ O 7 ;

[f ] v = (f _j ⁱ ) i,j . Then

1 = k(δ ⁻¹ f δ)v 1 k = kλ 1 (δ ⁻¹ f )v 1 k = kλ 1 7

X

i=1

1

λ _i f ₁ ⁱ v i k ⇒ f v 1 ∈ ker(δ − λ 1 I) Suppose f v _k ∈ ker(δ−λ k I) for all k 6 n. f v n+1 ∈ [f v 1 , . . . , f v _n ] ^⊥ , and thus f v _n+1 ∈ ker(δ − λ _n+1 I). By induction, all eigenspaces of δ are invariant under f . Therefore, f ∈ C _Aut(π) (δ), and δ ⁻¹ Aut(π)δ ∩ O ₇ ⊂ δ ⁻¹ C _Aut(π) (δ) δ ∩ O ₇ = C _Aut(π) (δ). The

converse is trivial. 

Summarizing Proposition 2.5, Proposition 2.8, Corollary 3.2, Corollary 4.4 and the above consideration, we obtain the following result concerning the automorphisms of real flexible division algebras:

Theorem 5.2 1. Let b > ^a

₂ ⁺¹ . Then Aut(A E (a, b)) = {I, (e ¹ , e 2 ) 7→ (e 1 , −e 2 )}

if a = 0 and Aut(A _E (a, b)) = {I} otherwise.

2. If (a, b) ∈ K ₂ = {(a, b) ∈ R ² | b > a = ¹ ₂ or a > b = ¹ ₂ }, then Aut(A _F (a, b)) = {I}.

3. If ab ∈ K ₃ = {(a, b) ∈ R ² | ab > ¹ ₄ , (a, b) 6∈ K ₂ }, then Aut(A F (a, b)) = {ϕ σ } σ∈S where

S =



 

 



 

 



{I, (1, 2)(3)} if a = b 6= −1 {I, (1)(2, 3)} if a = ^1−b _2b {I, (1, 3)(2)} if b = ^1−a _2a

S ₃ if a = b = −1

{I} otherwise

⊂ S ₃

4. Aut( _λ A) ∼ = SO 3 , if λ 6= 0 and A is a real flexible quadratic division algebra of dimension 4.

5. Aut( λ A) ∼ = C Aut(π) (δ) if λ 6= 0 and A ∼ = R(δ ^∗ π(δ ∧ δ)), where δ ∈ P ds(R ⁷ ) and π is a vector product on R ⁷ .

6. Aut(O γ ) = {κ A | A ∈ SU 3 }, if γ 6= 0.

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