Eight-Dimensional Real Quadratic Division Algebras

Ernst Dieterich

Abstract

Given a euclidean vector space V, a linear map η : V ∧ V → V is called dissident in case v, w, η(v∧w) are linearly independent whenever so are v, w ∈ V. The problem of classifying all real quadratic division algebras is reduced to the problem of classifying all eight-dimensional real quadratic division algebras, and further to the problem of classifying all dissident maps η : ℝ^{7} ∧ ℝ^{7} → ℝ^{7}. Should all of these satisfy η = επ for a vector product π on ℝ^{7} and a positive-definite endomorphism ε of ℝ^{7}, then the latter problem would be solved. This strong factorization property however, even though it does hold for all dissident maps in lower dimensions, is shown to fail in dimension 7. It is replaced by a weak factorization property for which a proof is announced. Evidence is given for the conjecture that even weak factorization will suffice to accomplish the complete classification of all eight-dimensional real quadratic division algebras.

1991 AMS Subject Classification: 15A21, 15A23, 17A05, 17A35, 17A45. Keywords: Vector product, weak vector product, dissident map, dissident triple, strong factorization, weak factorization.

1 Introduction

By a real algebra we mean a real vector space A, endowed with a bilinear multiplication A×A → A, (x, y) ↦ xy. A real division algebra is understood to be a real algebra A satisfying 0 < dim A < ∞ and having no zero divisors (i.e. xy = 0 only if x = 0 or y = 0). The old and challenging problem of classifying all real division algebras up to isomorphism has as yet admitted only partial solutions, the most significant of which we proceed to indicate.

The real numbers ℝ, the complex numbers ℂ, the quaternions ℍ (Hamilton 1843) and the octonions 𝕆 (Graves 1843, Cayley 1845) constitute the oldest known examples of real division algebras. Classical results of Frobenius [7] and Zorn [11] assert that {ℝ, ℂ, ℍ} classifies all associative real division algebras, whereas {ℝ, ℂ, ℍ, 𝕆} classifies all alternative real division algebras. Adams’ celebrated formula for the span of a sphere [1] implies that each real division algebra has dimension 1,2,4 or 8.

A real algebra A is called quadratic in case 0 < dim A, there exists an identity element 1 ∈ A and each x ∈ A satisfies an equation x^{2} = αx + β1 with real coefficients α, β. Every alternative real division algebra is quadratic. The classification of all quadratic real division algebras seems to be within reach. It is based on their intimate relationship with dissident triples which we proceed to recall. A dissident triple (V, ξ, η) consists of a finite-dimensional euclidean vector space V = (V, ⟨ ⟩), a linear form ξ : V ∧ V → ℝ and a dissident map η : V ∧ V → V. The category D of all dissident triples and the category Q of all quadratic real division algebras are related by a functor G : D → Q which is defined on objects by G(V, ξ, η) = ℝ×V, with multiplication (α, v)(β, w) = (αβ −⟨v, w⟩+ξ(v∧w), αw+βv+η(v∧w)), and on morphisms by G(σ) = 𝕀_{ℝ}×σ. ‘Frobenius Lemma’ [7],[8] and ‘Osborns Theorem’ [10] are predecessors of the following result.

Proposition 1.1 [3, pp. 15, 17], [5]. The functor G : D → Q is an equivalence of categories.

Intending to exploit this equivalence, let n ∈ {1, 2, 4, 8} and denote by Q_{n} the full subcategory of Q which is formed by all objects of dimension n. Likewise, set m = n − 1 and denote by D_{m} the full subcategory of D which is formed by all objects of dimension m. The (1,2,4,8)-Theorem and Proposition 1.1 reduce the classification problem for Q to the classification problem for D_{m}, where m ∈ {0, 1, 3, 7}. Clearly {(0, o, o)} classifies D_{0} and {(ℝ, o, o)} classifies D_{1}, so {ℝ} classifies Q_{1} and {ℂ} classifies Q_{2}. In case m ∈ {3, 7} we are faced with the key subproblem of classifying all dissident maps ℝ^{m} ∧ ℝ^{m} → ℝ^{m} up to O(ℝ^{m})-action. It is partially solved by the following result, taking into account that the set of all vector products on ℝ^{m} forms one orbit under this group action.

Proposition 1.2 [3, p. 19], [5]. Let V be a euclidean vector space of dimension m ∈ {3, 7}.

(i) If π : V ∧ V → V is a vector product and ε : V → V is a definite endomorphism, then επ : V ∧ V → V is a dissident map. (ii) If π, π′ : V ∧ V → V are vector products and ε, ε′ : V → V are definite endomorphisms, then επ = ε′π′ iff (ε, π) = (ε′, π′) or (ε, π) = (−ε′, −π′). (iii) If m = 3 and π : V ∧ V → V is a fixed vector product, then ε ↦ επ is a bijection between the set of all definite endomorphisms ε : V → V and the set of all dissident maps V ∧ V → V.

If m = 3, choose your favourite vector product π : ℝ^{3} ∧ ℝ^{3} → ℝ^{3}. For each x ∈ ℝ^{3}, denote by A_{x} the antisymmetric matrix representing the endomorphism π(x∧?) : ℝ^{3} → ℝ^{3} in the standard basis. Let K_{3} = ℝ^{3} × ℝ^{3} × T , where T = {d ∈ ℝ^{3} | 0 < d_{1} ≤ d_{2} ≤ d_{3}}. For each d ∈ T , denote by D_{d} the diagonal matrix with diagonal sequence d. Define an equivalence relation on K_{3} by (x, y, d) ∼ (x′, y′, d′) iff d = d′ and (Sx, Sy) = (x′, y′) for some S ∈ SO_{3}(ℝ) satisfying SD_{d}S^{T} = D_{d}. Now we can formulate a first consequence of Propositions 1.1 and 1.2, providing a complete description of the isoclasses of Q_{4} in terms of equivalence classes of configurations in ℝ^{3}, formed by a pair of points and an ellipsoid in normal position.

Proposition 1.3 [4, p. 944]. The map Φ_{3} : K_{3} → Q_{4}, (x, y, d) ↦ ℝ × ℝ^{3}, (α, v)(β, w) = (αβ − v^{T} w + v^{T} A_{x}w , αw + βv + (A_{y} + D_{d})π(v ∧ w)), induces a bijection \overline{Φ}_{3} : K_{3}/∼ ˜→ Q_{4}/≃ .

A cross-section C for K_{3}/∼ is given in [2, pp. 17-18]. From Proposition 1.3 we infer that Φ_{3}(C) classifies Q_{4}.

If m = 7, choose any vector product π : ℝ^{7} ∧ ℝ^{7} → ℝ^{7}. Let K_{7} be the set of all matrix pairs (X, E) ∈ ℝ^{7×7} × ℝ^{7×7} such that X is antisymmetric and E is positive-definite. Define an equivalence relation on K_{7} by (X, E) ∼ (X′, E′) iff (SXS^{T} , SES^{T}) = (X′, E′) for some S ∈ O_{7}(ℝ) stabilizing π. Imitating as far as possible the reasoning applied for m = 3, one obtains the following construction of isoclasses of Q_{8}.

Proposition 1.4 [3, p. 20],[5]. The map Φ_{7} : K_{7} → Q_{8}, (X, E) ↦ ℝ × ℝ^{7}, (α, v)(β, w) = (αβ − v^{T} w + v^{T} Xw , αw + βv + Eπ(v ∧ w)), induces an injection \overline{Φ}_{7} : K_{7}/∼ ↪ Q_{8}/≃ .

2 On the factorization of dissident maps

The last piece that is missing to make Propositions 1.3 and 1.4 a complete classification of Q is the surjectivity of \overline{Φ}_{7}. Although the argument for the surjectivity of \overline{Φ}_{3}, namely the bijectivity of π : ℝ^{3} ∧ ℝ^{3} → ℝ^{3}, does not generalize to \overline{Φ}_{7}, one still might hope that \overline{Φ}_{7} is surjective for other reasons. In what follows we shall outline an argument proving that this hope fails to be true. And we shall indicate a first step in closing the gap that is left open by the non-surjectivity of \overline{Φ}_{7}.

Given a finite-dimensional euclidean vector space V = (V, ⟨ ⟩), we denote by 𝔼(V) the set of all two-dimensional subspaces of V. Each linear map φ : V ∧ V → V induces a map

φ_{𝔼} : 𝔼(V) → ℝ_{≥0} , φ_{𝔼}(P) = ⟨φ(v ∧ w), φ(v ∧ w)⟩ ,

where (v, w) is any orthonormal basis for P ∈ 𝔼(V). Apart from being dissident, or even a vector product, we now introduce two more properties for linear maps φ : V ∧ V → V. We call φ plane-constant iff φ_{𝔼} is constant with positive value. We call φ a weak vector product iff it satisfies ⟨φ(u ∧ v), w⟩ = ⟨u, φ(v ∧ w)⟩ for all (u, v, w) ∈ V^{3} and φ(v ∧ w) ≠ 0 for all non-proportional pairs (v, w) ∈ V^{2}. Note that each vector product is a plane-constant weak vector product, and each weak vector product is dissident.

Whereas the map Φ_{7} depends on a chosen vector product on ℝ^{7}, the set of isoclasses \overline{Φ}_{7}(K_{7}/∼) no longer does it. The surjectivity of \overline{Φ}_{7} can now be characterized as follows.

Proposition 2.1 [6]. The following assertions are equivalent:

(i) The injective map \overline{Φ}_{7} : K_{7}/∼ ↪ Q_{8}/≃ is surjective. (ii) Every dissident map η : ℝ^{7} ∧ ℝ^{7} → ℝ^{7} admits strong factorization. (iii) Every weak vector product ϱ : ℝ^{7} ∧ ℝ^{7} → ℝ^{7} is plane-constant.

The group GL(ℝ^{7}) acts on the set Wec(ℝ^{7}) of all weak vector products on ℝ^{7} by

Wec(ℝ^{7}) × GL(ℝ^{7}) → Wec(ℝ^{7}), ϱ · ε = ε^{∗}ϱ(ε ∧ ε),

where ε^{∗} denotes the adjoint of ε. Restricting this group action to the set Vec(ℝ^{7}) of all vector products on ℝ^{7}, we obtain the map

Vec(ℝ^{7}) × GL(ℝ^{7}) → Wec(ℝ^{7}), π · ε = ε^{∗}π(ε ∧ ε)

which we view as a means to construct weak vector products out of a given vector product. Now we apply this construction to the ‘standard’ vector product π induced from 𝕆 on its purely imaginary hyperplane (cf. [9]) and, accordingly, having the multiplication table

\begin{matrix}π & e_{1} & e_{2} & e_{3} & e_{4} & e_{5} & e_{6} & e_{7} \\ e_{1} & 0 & e_{3} & −e_{2} & e_{5} & −e_{4} & −e_{7} & e_{6} \\ e_{2} & −e_{3} & 0 & e_{1} & e_{6} & e_{7} & −e_{4} & −e_{5} \\ e_{3} & e_{2} & −e_{1} & 0 & e_{7} & −e_{6} & e_{5} & −e_{4} \\ e_{4} & −e_{5} & −e_{6} & −e_{7} & 0 & e_{1} & e_{2} & e_{3} \\ e_{5} & e_{4} & −e_{7} & e_{6} & −e_{1} & 0 & −e_{3} & e_{2} \\ e_{6} & e_{7} & e_{4} & −e_{5} & −e_{2} & e_{3} & 0 & −e_{1} \\ e_{7} & −e_{6} & e_{5} & e_{4} & −e_{3} & −e_{2} & e_{1} & 0 \end{matrix}

in the standard basis e_{1}, …, e_{7} of ℝ^{7}. Defining ε ∈ GL(ℝ^{7}) by ε(e_{i}) = ε_{i}e_{i} for all i ∈ {1, …, 7}, where 0 < ε_{1} < … < ε_{7} is any chosen positive ascending septuple, we obtain the weak vector product ϱ = π ·ε. Its induced map ϱ_{𝔼} satisfies strict inequalities such as e.g.

ϱ_{𝔼}(ℝe_{1} ⊕ ℝe_{2}) = (ε_{1}ε_{2}ε_{3})^{2} < (ε_{1}ε_{4}ε_{5})^{2} = ϱ_{𝔼}(ℝe_{1} ⊕ ℝe_{4}) .

Hence ϱ is not plane-constant. From Proposition 2.1 we infer that \overline{Φ}_{7} is not surjective. Moreover, not every dissident map on ℝ^{7} admits strong factorization. However, combining ideas from real algebraic geometry with matrix formulae drawn from linear statistical inference theory, we are able to prove that each dissident map on ℝ^{7} does admit weak factorization, in the following sense.

Proposition 2.2 [6]. For each dissident map η : ℝ^{7} ∧ℝ^{7} → ℝ^{7} there exist a weak vector product ϱ : ℝ^{7} ∧ ℝ^{7} → ℝ^{7} and a positive-definite endomorphism ε : ℝ^{7} → ℝ^{7} such that η = εϱ. Weak factorization reduces the problem of classifying Q_{8} essentially to the problem of classifying all weak vector products on ℝ^{7} up to O(ℝ^{7})-action. The very same map Vec(ℝ^{7}) × GL(ℝ^{7}) → Wec(ℝ^{7}) which just destroyed our hope that Φ_{7} might completely classify Q_{8} will be an important tool for the solution of this remaining problem.

References

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[3] Dieterich, E.: Dissident algebras, Colloquium Mathematicum 82, 13-23 (1999).

[4] Dieterich, E.: Real quadratic division algebras, Commun. in Algebra 28(2), 941-947 (2000).

[5] Dieterich, E.: Quadratic division algebras revisited (Remarks on an article by J.M. Osborn), Proc. Amer. Math. Soc. 128, 3159-3166 (2000).

[6] Dieterich, E.: On the factorization of dissident maps. To appear.

[7] Frobenius, F.G.: Über lineare Substitutionen und bilineare Formen, Journal für die reine und angewandte Mathematik 84, 1-63 (1878).

[8] Koecher, M. und Remmert, R.: Isomorphiesätze von Frobenius, Hopf und Gelfand-Mazur. Zahlen, Springer-Lehrbuch, 3. Auflage, 182-204 (1992).

[9] Koecher, M. und Remmert, R.: Kompositionsalgebren. Satz von Hurwitz. Vektorprodukt-Algebren. Zahlen, Springer-Lehrbuch, 3. Auflage, 219-232 (1992).

[10] Osborn, J.M.: Quadratic division algebras, Trans. Amer. Math. Soc. 105, 202-221 (1962).

[11] Zorn, M.: Theorie der alternativen Ringe, Abh. Math. Sem. Hamburg 8, 123-147 (1931).

Matematiska institutionen, Uppsala universitet, Box 480, S-751 06 Uppsala, Sweden E-mail: Ernst.Dieterich@math.uu.se

[AMA - Algebra Montpellier Announcements - 01-2000][October, 2000]
Received September 2000.