A QUICK PROOF OF THE 1, 2, 4, 8 THEOREM

M. G. MAHMOUDI

In this note we provide a quick proof of a well-known theorem on the possible dimensions of composition algebras over an arbitrary field. This theorem has an old history originated from a theorem of A. Hurwitz published in 1898 about the compositions of quadratic forms. See [3] or [6] for further details and references.

Our proof is inspired by an elegant proof of Hurwitz’s theorem due to C. Chevalley given in [1, §2.8], but is different from it, and to the best of our knowledge has not previously appeared in the literature. Also an auxiliary map which is crucial in the proof, is already present in the literature to obtain some versions of the Principle of Triality (see [4, (4.4)], [2, §1]). Apart from using Clifford algebras, the main ingredient of the proof presented here, is to use the ‘centralizer theorem’ in the theory of central simple algebras. The centralizer theorem implies in particular that the dimension of a simple subalgebra of a central simple algebra A divides that of A, see [5, §1].

To render the text more self-contained, we recall some basic properties of composition algebras; these are all standard and can be found for instance in [6, Ch. 1] or [5, §33].

A finite dimensional algebra C (not necessarily associative) with the unity e over a field F is called a composition algebra, if there exists a nondegenerate quadratic form (i.e., its associated bilinear form is nondegenerate, see the comments below) N : C → F such that

(1)  N(xy) = N(x)N(y), for all x, y ∈ C.

Let ⟨x, y⟩ = N(x + y) − N(x) − N(y) be the bilinear form associated to N. The bilinearity of ⟨, ⟩ and the relation (1) imply

(2)  ⟨ax, ay⟩ = N(a)⟨x, y⟩, ⟨xa, ya⟩ = N(a)⟨x, y⟩, for all a, x, y ∈ C.

By replacing a by b + c in one of the relations (2), we obtain

(3)  ⟨xb, yc⟩ + ⟨xc, yb⟩ = ⟨x, y⟩⟨b, c⟩, for all x, y, b, c ∈ C.

We show that the above relations imply that the following identity holds in C:

(4)  x^{2} − ⟨x, e⟩x + N(x)e = 0.

As ⟨, ⟩ is nondegenerate, in order to prove (4), it is sufficient to show that ⟨x^{2} − ⟨x, e⟩x + N(x)e, y⟩ = 0 for all y ∈ C. This relation reduces to

⟨x^{2}, y⟩ + ⟨x, e⟩⟨x, y⟩ + ⟨x, yx⟩ = 0,

which is valid in virtue of (3). Also the ‘bar involution’¯ : C → C, is defined as

x̄ = ⟨x, e⟩e − x.

The relation (4) implies that the following identity holds:

(5)  xx̄ = x̄x = N(x)e.

We also have

(6)  ⟨cx, y⟩ = ⟨x, c̄y⟩, for all c, x, y ∈ C.

To obtain the above relation write ⟨x, c̄y⟩ = ⟨x, (⟨c, e⟩e − c)y⟩ = ⟨c, e⟩⟨x, y⟩ − ⟨x, cy⟩ = ⟨cx, y⟩, where in the last step, the relation (3) was used. Using (6) we obtain ⟨c(c̄x), y⟩ = ⟨c̄x, c̄y⟩ = N(c̄)⟨x, y⟩ = ⟨N(c)x, y⟩ for all y ∈ C, thus

(7)  c(c̄x) = N(c)x = (cc̄)x, for all c, x ∈ C.

Note that (7) is equivalent to

(8)  c(cx) = (cc)x, for all c, x ∈ C.

This alternative law is a special case of the Moufang identities. An algebra in which the identity (8) holds is called a left alternative algebra. In presence of an involution every left alternative algebra is also a right alternative algebra.

The Clifford algebra of (C, N) is denoted by Cl(C, N).

Theorem 1. The possible dimensions of a composition algebra C are 1, 2, 4 or 8.

Proof. We may assume that n = dim C > 1. Consider the map φ : C → End(C⊕C) defined by φ(c)(x, y) = (cy, c̄x), c ∈ C, x, y ∈ C. Thanks to (7) we have φ(c)^{2} = N(c) · id. So φ can be extended to an F -algebra homomorphism

Φ : Cl(C, N) → End(C ⊕ C).

Note that dim C is even (compare dim C with the rank of C over K = F [a] where a ∈ e^{⊥} is an anisotropic vector), so Cl(C, N) is central simple and Φ is injective. By centralizer theorem dim Cl(C, N) divides dim End(C ⊕ C), so n = 1, 2, 4, 8. □

Comments. (i) The above proof just determines the possible dimensions of C and does not give insight into the structure of C and its relations with the Cayley-Dickson algebras. (ii) There exist purely inseparable extensions C/F of degree 2^{n} with N(x) := x^{2} ∈ F for all x ∈ C. The form N is nondefective, so N should be nondegenerate in traditional sense. This is one of the reasons why the authors of [6] have preferred to call a quadratic form nondegenerate if its associated bilinear form is nondegenerate. (iii) In [5], a quadratic form N : V → F is called nondegenerate if either V^{⊥} = {0} or dim V^{⊥} = 1 and N(V^{⊥}) ≠ 0. The second case only makes sense if charF = 2 and V is odd dimensional. By [5, (33.4)], if N is arbitrary and the identity (1) holds (in other words N is multiplicative), then either N(C^{⊥}) = {0} or charF = 2 and C = C^{⊥}. Thus if we are in the second case of this notion of nondegeneration, every composition algebra is necessarily one dimensional.

Acknowledgments. The support from Sharif University of Technology is gratefully acknowledged.

References

[1] C. Chevalley, Collected works. Vol. 2: The algebraic theory of spinors and Clifford algebras, Edited by Pierre Cartier and Catherine Chevalley, Springer, Berlin, 1997.

[2] A. Elduque, On triality and automorphisms and derivations of composition algebras, Linear Algebra Appl. 314, No. 1-3 (2000) 49-74.

[3] N. Jacobson, Composition algebras and their automorphisms, Rend. Circ. Mat. Palermo, II. Ser. 7 (1958) 55-80.

[4] M.-A. Knus, R. Parimala, R. Sridharan, On compositions and triality, J. reine angew. Math. 457 (1994) 45-70.

[5] M.-A. Knus, A. S. Merkurjev, M. Rost, J.-P. Tignol, The book of involutions, American Mathematical Society Colloquium Publications, 44. American Mathematical Society, Providence, RI, 1998.

[6] T. A. Springer, F. D. Veldkamp, Octonions, Jordan algebras and exceptional groups,

Springer Monographs in Mathematics, Springer, Berlin, 2000.

M. G. Mahmoudi, mmahmoudi@sharif.ir, Department of Mathematical Sciences, Sharif University of Technology, P. O. Box 11155-9415, Tehran, Iran. Fax: (+98) (21) 6616-5117