THE HERMITIAN LEVEL OF COMPOSITION ALGEBRAS

S. PUMPLÜN AND T. UNGER

Abstract. The hermitian level of composition algebras with involution over a ring is studied. In particular, it is shown that the hermitian level of a composition algebra with standard involution over a semilocal ring, where two is invertible, is always a power of two when finite. Furthermore, any power of two can occur as the hermitian level of a composition algebra with nonstandard involution. Some bounds are obtained for the hermitian level of a composition algebra with involution of the second kind.

1. Introduction

Let R be a unital, commutative, associative ring, equipped with a non-trivial involution (i.e. an anti-automorphism of period 2). For x ∈ R, we call \overline{x}x a hermitian square. The smallest positive integer m such that −1 is a sum of m hermitian squares in R is called the hermitian level of (R, ), denoted S_{h}(R, ). If no such m exists, we set S_{h}(R, ) = ∞. Note that S_{h}(R, ) depends on the involution and not just on the ring R and also that if = id_{R}, then S_{h}(R, ) = s(R) is the usual level.

In [7], Lewis showed that there exist commutative rings with non-trivial involution having any positive integer as hermitian level. He also showed that the hermitian level, if finite, is a power of two for fields K of characteristic not two with non-trivial involution and for quaternion division algebras D over fields of characteristic not two with the standard involution.

K and D are exactly the non-split composition algebras of dimension 2 and 4 over a suitable field k. It is well-known that composition algebras only exist in dimension 1, 2, 4 and 8. An 8-dimensional composition algebra C is called an octonion (or Cayley) algebra and, although it is not associative, it retains the most pleasing property of K and D: it comes equipped with a non-degenerate norm form with values in k, which is multiplicative.

We investigate the hermitian level of composition algebras C with involution over a ring R, extending the results of Lewis [7] for the standard involution and some of the results of Serhir [14] for the non-standard involution on C (which we introduce here in a more general context). In particular, we show that over semilocal rings where 2 is invertible, the hermitian level of (C, ) is a power of two if it is finite and that all powers of two can occur as the hermitian level of (C, ). Finally, we obtain some bounds on the hermitian level of C, equipped with an involution of the second kind.

Date: 21 March 2002.

2000 Mathematics Subject Classification. 17A75, 16W10, 11E25.

Our methods are mostly quadratic form theoretic in nature and usually adaptations of Lewis’s arguments to the more general setting of rings. We refer to the standard works [1], [6] and [12] for background reading on quadratic forms over rings, algebras with involution and nonassociative algebras respectively.

2. Preliminaries

2.1. Composition Algebras with Involution. Let R be a unital commutative associative ring, and A a unital nonassociative R-algebra which is finitely generated projective of (constant) rank > 0 as an R-module. We will simply call such an algebra A an “R-algebra” in the remainder of this paper.

Associativity in A is measured by the associator [x, y, z] = (xy)z − x(yz), commutativity by the commutator [x, y] = xy − yx. Define the nucleus of A by

Nuc(A) = {x ∈ A | [x, A, A] = [A, x, A] = [A, A, x]}

and the commuter of A by

Comm(A) = {x ∈ A | [x, A] = 0}.

An R-algebra C is called a composition algebra, if it carries a quadratic form n: C → R satisfying the following two conditions:

(i) Its induced symmetric bilinear form n(x, y) := n(x + y) − n(x) − n(y) is nondegenerate, i.e. determines an R-module isomorphism C ⟶^{∼} C^{∨} = Hom_{R}(C, R).

(ii) n permits composition, that is n(xy) = n(x)n(y) for all x, y ∈ C.

A composition algebra over R is split if it contains a composition subalgebra, isomorphic to R ⊕ R. (See [9] for an explicit description of split composition algebras.)

An R-algebra A is called quadratic in case there exists a quadratic form n: A → R such that

n(1_{A}) = 1 and x^{2} − n(1_{A}, x)x + n(x)1_{A} = 0

for all x ∈ A. The form n is uniquely determined and called the norm of the quadratic algebra A.

A nonassociative algebra is called alternative if its associator [x, y, z] is alternating. Composition algebras are quadratic alternative algebras. More precisely, a quadratic form n of the composition algebra satisfying (i) and (ii) above agrees with its norm as a quadratic algebra and thus is unique. It is called the norm of the composition algebra C and is sometimes also denoted by n_{C}.

A quadratic alternative algebra is a composition algebra if and only if its norm is nondegenerate [8, 4.6]. Composition algebras only exist in ranks 1, 2, 4 or 8. Those of rank 2 are exactly the quadratic étale R-algebras, those of rank 4 exactly the well-known quaternion algebras. The ones of rank 8 are also called octonion algebras.

Let τ be an involution on A, i.e. a map τ : A → A satisfying

τ(x + y) = τ(x) + τ(y), τ(xy) = τ(y)τ(x), and τ^{2}(x) = x

for all x, y ∈ A.

If 2 is an invertible element in R, we have A = Skew(A, τ) ⊕ Sym(A, τ) with

Skew(A, τ) := {x ∈ A | τ(x) = −x}

the set of skew-symmetric elements and

Sym(A, τ) := {x ∈ A | τ(x) = x}

the set of symmetric elements in A with respect to τ. An involution is called scalar if all the norms τ(x)x are scalars in R, and hence by linearization all traces τ(x) + x are scalars in R, too. Note that for a scalar involution, n_{A}(x) = τ(x)x resp. t_{A}(x) = τ(x) + x is a quadratic resp. a linear form on A, whenever a1_{A} = 0 implies a = 0, for every a ∈ R [8, p. 86].

An involution τ on A is called of the first kind if τ|_{R} = id. Let S be a quadratic étale R-algebra with standard involution s_{0}, and let A be an algebra over S. An involution on A whose restriction to S is the standard involution is called of the second kind.

Given an R-algebra A together with an involution τ, an element in A of the form τ(x)x with x ∈ A is called a hermitian square in A. Following [7] the hermitian level of A is the least integer m such that

−1 = τ(x_{1})x_{1} + ⋯ + τ(x_{m})x_{m}.

We write S_{h}(A, τ) for the hermitian level and define S_{h}(A, τ) = ∞ if −1 cannot be written as a sum of hermitian squares in (A, τ). Obviously, the hermitian level of an algebra depends on the particular involution and not just on the algebra.

Let a_{i} ∈ R^{×} (1 ≤ i ≤ m) be invertible. We define a map h : A^{m} × A^{m} → A by

h(x, y) := a_{1}τ(x_{1})y_{1} + ⋯ + a_{m}τ(x_{m})y_{m}

for x = (x_{1}, …, x_{m}), y = (y_{1}, ⋯, y_{m}) ∈ A^{m}. This map is biadditive and satisfies

h(xα, yβ) = a_{1}τ(x_{1}α)(y_{1}β) + ⋯ + a_{m}τ(x_{m}α)(y_{m}β) = τ(α)h(x, y)β

provided that R1_{A} ⊂ Comm(A) ∩ Nuc(A). The map h is called isotropic if there are x_{1}, …, x_{m} ∈ A with at least one invertible x_{i} ∈ A^{×}, such that

∑_{i=1}^{m} a_{i}τ(x_{i})x_{i} = 0,

otherwise it is called anisotropic. Write m × ⟨1⟩ for the map h : A^{m} × A^{m} → A, represented by the quadratic form ⟨1, …, 1⟩ = m×⟨1⟩, i.e. for h(x, y) = τ(x_{1})y_{1}+ ⋯ + τ(x_{m})y_{m}.

A composition algebra C comes endowed with a standard involution , given by \overline{x} = t(x)1_{C} − x, where t: C → R is the trace map given by t(x) := n(1_{C}, x). This involution is scalar.

2.2. The Cayley-Dickson Doubling Process. The Cayley-Dickson doubling process is a well-known way to construct new algebras with scalar involution out of a given one with such an involution. Let A be an R-algebra with involution ∗ and let μ ∈ R be such that μx = 0 implies x = 0 in A. Then the R-module A ⊕ A becomes a unital algebra via the multiplication

(u, v)(u′, v′) = (uu′ + μv′^{∗}v, v′u + vu′^{∗})

for u, u′, v, v′ ∈ A, with involution

(u, v)^{∗} = (u^{∗}, −v).

It is called the (classical) Cayley-Dickson doubling of A, and denoted by Cay(A, μ). The involution ∗ is a scalar involution on Cay(A, μ) if and only if ∗ is a scalar involution on A, with norm

n_{Cay(A,μ)}((u, v)) = n_{A}(u) − μn_{A}(v).

The Cayley-Dickson doubling process depends on the scalar μ only up to an invertible square. By repeated application of the Cayley-Dickson doubling process, starting from a composition algebra C over R, we obtain either again a composition algebra (if the rank of the new algebra is less than or equal to 8), or a generalized Cayley-Dickson algebra of rank 2^{m} rank(C) ≥ 16. The latter are no longer alternative, but still flexible (i.e. (xy)x = x(yx) holds for all x, y in the algebra) with a scalar involution (cf. [8]).

When iterating the Cayley-Dickson doubling process, we use the shorthand notation

Cay(A, μ, ν) := Cay(Cay(A, μ), ν).

Over fields, the classical Cayley-Dickson process indeed generates all possible composition algebras. Over rings, a more general version is required, which still does not always yield all possible composition algebras, only those containing a composition subalgebra of half its rank. This “generalized” Cayley-Dickson doubling process is due to Petersson [9]. Its description is quite technical. Here is the main result:

Let D be a composition algebra over R of rank ≤ 4 with standard involution . Let P be a finitely generated projective right D-module of rank one, carrying a nondegenerate hermitian form h: P × P → D. Define N: P → D by N(w) := h(w, w) for w ∈ P. For each μ ∈ R, the R-module

A = D ⊕ P

becomes a new R-algebra via the multiplication

(u, w)(u′, w′) = (uu′ + μh(w′, w), w′ · u + w · \overline{u}′)

for u, u′ ∈ D, w, w′ ∈ P, and where · denotes the right D-module structure of P. The algebra A constructed above is denoted by Cay(D, P, μN) := Cay(D, P, μh). D itself is canonically a (free) right D-module of rank one and “norm one” (cf. [9]). Any norm on D is similar to n_{D} (resp. any nondegenerate hermitian form h: D × D → D is similar to the canonical form given by the involution, i.e., by (w, w′) ↦ \overline{w}′w). In this special case the “classical” Cayley-Dickson doubling process Cay(D, μ) := Cay(D, D, μn_{D}) with μ ∈ D is obtained.

3. The Standard Involution

When investigating the hermitian level, one of the easiest classes of nonassociative algebras to look at are algebras A with a scalar involution, since for these the hermitian square τ(x)x is an element in R, and the map x → τ(x)x a quadratic form, provided that the ring R is isomorphic to a subring of A (cf. [8]). We start our investigation by looking at composition algebras with standard involution .

If k is a field of characteristic not two, and C a split composition algebra over k with standard involution , then it is easy to see that −1 = \overline{x}x for an element x ∈ {y ∈ C | t(y) = 0} = C_{0}, i.e. for a pure quaternion (resp. octonion) in case C is a quaternion (resp. octonion) algebra. More generally, we obtain:

Lemma 3.1. Let R be a ring with 2 ∈ R^{×}, and C a split composition algebra over R with standard involution , then S_{h}(C, ) = 1.

Proof. C is split if and only if C contains the split quadratic étale algebra R ⊕ R with hyperbolic norm n ≃ ⟨1, −1⟩, in particular, n_{C} ≃ ⟨1, −1⟩ ⊥ ⋯ which implies that n_{C}(x) = \overline{x}x = −1 for a suitable element x ∈ C.

Note that any quaternion algebra over R with norm n_{C} ≃ ⟨1, −1⟩ ⊥ ⋯ splits, but that this is not necessarily so for an octonion algebra.

Lemma 3.2. Let R be a ring and let C be a composition algebra over R with standard involution .

(1) If S_{h}(C, ) = m, then the form (m + 1) × n_{C} is isotropic. Moreover, if m × n_{C} already is isotropic, then every isotropic vector (x_{1}, …, x_{m}) satisfies n_{C}(x_{i}) = \overline{x}_{i}x_{i} ∉ R^{×} (or, equivalently, x_{i} ∉ C^{×}), for all i, 1 ≤ i ≤ m.

(2) Let m × n_{C} be anisotropic and let (x_{1}, …, x_{m+1}) be an isotropic vector of (m+1)×n_{C} such that x_{i} ∈ C^{×} for all i, 1 ≤ i ≤ (m+1), then S_{h}(C, ) = m.

Proof. (1) Let m be the smallest integer such that −1 = \overline{x}_{1}x_{1} +⋯+\overline{x}_{m}x_{m}. Then 0 = 1 + \overline{x}_{1}x_{1} + ⋯ + \overline{x}_{m}x_{m}, hence (m + 1) × n_{C} is isotropic. In case m × n_{C} is isotropic, and there exists an isotropic vector (x_{1}, …, x_{m}) where x_{i} is invertible in C, for some i, 1 ≤ i ≤ m, then 0 = \overline{x}_{1}x_{1} + ⋯ + \overline{x}_{m}x_{m}. Without loss of generality assume x_{1} ∈ C^{×}. Then also \overline{x}_{1} ∈ C^{×} and 0 = 1 + (\overline{x_{1}^{−1}x_{2}})(x_{1}^{−1}x_{2}) + ⋯ + (\overline{x_{1}^{−1}x_{m}})(x_{1}^{−1}x_{m}), contradicting the minimality of m.

(2) Let (m + 1) × n_{C} be isotropic. Then as above this implies S_{h}(C, ) ≤ m, provided that there exists an isotropic vector (x_{1}, …, x_{m+1}) such that x_{i} is invertible in C, for some i, 1 ≤ i ≤ m + 1. Let s = S_{h}(C, ), then −1 = \overline{x}_{1}x_{1} + ⋯ + \overline{x}_{s}x_{s} and (s + 1) × n_{C} is isotropic. Since m × n_{C} is anisotropic, we must have m < s + 1 and conclude that S_{h}(C, ) = m.

Note that for any algebra with involution (A, τ), S_{h}(A, τ) = m always implies that the “pseudo-hermitian form” (m + 1) × ⟨1⟩, as defined in 2.1, is isotropic. However, the multiplicativity of the norm and the fact that the standard involution of a composition algebra is scalar are crucial in the converse of the above lemma.

In particular, for semilocal rings we obtain a generalization of [7, 1.1] (applying [1, III.(5.2)(iii)] as for instance in [1, III.(5.3), p. 87]):

Corollary 3.3.

(1) Let be the standard involution on a composition algebra C of rank greater than 1 over a semilocal ring R with 2 ∈ R^{×}. Let m ≥ 2 and assume that either m is even, or that |R/𝔪| ≥ 3, for all 𝔪 ∈ max(R). Then S_{h}(C, ) = m if and only if m × n_{C} = ⟨1, …, 1⟩ ⊗ n_{C} (m-times) is anisotropic, but (m + 1) × n_{C} is isotropic.

(2) Let be the standard involution on a composition division algebra C over a field k of char(k) ≠ 2. Then S_{h}(C, ) = m if and only if m × n_{C} = ⟨1, …, 1⟩ ⊗ n_{C} (m-times) is anisotropic, but (m + 1) × n_{C} is isotropic.

Lemma 3.4 (cf. [7, 1.2]). Let C be a composition division algebra over a field k of char(k) ≠ 2 with standard involution . The following are equivalent:

(1) Zero is a nontrivial sum of hermitian squares, i.e. there exists an integer m such that m × n_{C} is an isotropic quadratic form.

(2) −1 is a sum of hermitian squares.

(3) Each symmetric element of C is a sum of hermitian squares.

Proof. The above corollary implies the equivalence of (1) and (2).

(3) ⇒ (2) is obvious, since −1 ∈ Sym(C, ).

(1) ⇒ (3): Let 0 = \overline{x}_{1}x_{1} + ⋯ + \overline{x}_{m}x_{m} = n_{C}(x_{1}) + ⋯ + n_{C}(x_{m}) with x_{i} ∈ C not all zero. Then m × n_{C} is an isotropic k-quadratic form, hence universal, and represents all invertible elements a ∈ Sym(C, ). That is, for any a ∈ k^{×} there are x_{i} ∈ C such that a = n_{C}(x_{1}) + ⋯ + n_{C}(x_{m}).

The conclusion that (3) implies (2) implies (1) in the above proof also holds for composition algebras over rings R where 2 is invertible. The proof of (1) implies (3) holds provided that the following is true: If the R-quadratic form ⟨1, …, 1⟩ ⊗ n_{C} is isotropic it contains a hyperbolic plane.

The following generalizes [7, 1.5].

Proposition 3.5. Let C be a composition algebra over a semilocal ring R with 2 ∈ R^{×}, such that n_{C} = ⟨⟨a, b⟩⟩ or n_{C} = ⟨⟨a, b, c⟩⟩ is a Pfister form (this is always the case if C has rank ≤ 4). If S_{h}(C, ) is finite then S_{h}(C, ) is a power of 2. More precisely, then S_{h}(C, ) = 2^{−1}d with d the order of the norm n_{C} in W (R), the Witt group of non-singular quadratic forms over R.

Proof. Assume that S_{h}(C, ) = m is finite. Then −1 = \overline{x}_{1}x_{1} + ⋯ + \overline{x}_{m}x_{m} = n_{C}(x_{1}) + ⋯ + n_{C}(x_{m}) and (m + 1) × n_{C} is isotropic. Pick s such that 2^{s} < m + 1 ≤ 2^{s+1}. Then 2^{s+1} × n_{C} ≅ ⟨1, …, 1⟩ ⊗ n_{C} is an isotropic Pfister form, thus hyperbolic [1, IV.(3.2)]. Therefore

2^{s+1} × n_{C} ≅ ⟨⟨−1, …, −1⟩⟩ ≅ ⟨1, −1, …, 1, −1⟩ ⊗ n_{C}

and by Witt cancellation, ⟨−1, …, −1⟩ ⊗ n_{C} ≅ ⟨1, …, 1⟩ ⊗ n_{C}. It follows that −1 ∈ D(2^{s} × n_{C}), i.e. −1 is represented by 2^{s} × n_{C}, and so m = 2^{s}.

Moreover, obviously n_{C} is a torsion space whose order d in W (R) is less than or equal to 2^{s+1}. Now d must be a power of 2 [1, (6.1), p. 143], hence even d = 2^{s+1}, since 2^{s} × n_{C} ∼ 0 implies that −1 is a sum of 2^{s−1} < m hermitian squares in (C, ), a contradiction. We conclude that m = 2^{−1}d.

Corollary 3.6. Let C be a composition (division) algebra over a field k of char(k) ≠ 2. If S_{h}(C, ) is finite then S_{h}(C, ) is a power of 2. More precisely, then S_{h}(C, ) = 2^{−1}d with d the order of the norm n_{C} in W (k).

Even more general, let A be a generalized Cayley-Dickson algebra of rank 2^{r} over a semilocal ring R with 2 invertible in the sense of [8, (6.2)], obtained from R1 by a repeated application of the (classical) Cayley-Dickson doubling process with scalar involution ∗ as described in [8, (6.3)]. Then S_{h}(A, ∗) also is a power of two if it is finite, by the same argument as in the above proof.

An immediate consequence of the last proposition is the fact that any prescribed power of two occurs as the level of an octonion division algebra with standard involution over a field (cf. [7, Comment 2, p.470]). Let

k = ℝ(x_{1}, …, x_{m})(\sqrt{− ∑_{i=1}^{m} x_{i}^{2}})

be a field of level s(k) = S_{h}(k, id) = 2^{s}. If s(k) ≥ 8, then the octonion algebra Cay(k, −1, −1, −1) with standard involution has hermitian level 2^{s−3}.

Recall that the level s(R) of a semilocal ring R, where 2 is invertible, is either infinite or a power of two [1, (A.3), p. 177]. The following analog to [7, (1.7)] holds in the setting of semilocal rings (it even holds for a generalized Cayley-Dickson algebra obtained from R1 with scalar involution ∗):

Proposition 3.7 ([4, (1.5)], [1, (2.4), p. 95]). Let R be a semilocal ring where 2 is invertible. Let C be a composition algebra over R with standard involution whose norm n_{C} is a Pfister form. Let t be a positive integer. The set of invertible values represented by the quadratic form n_{C} ⊗ ⟨⟨1, …, 1⟩⟩ = 2^{t} × n_{C} over R is a group.

However, we can do better than this:

Proposition 3.8 ([13, 5.6]). Let R be a semilocal ring where 2 is invertible. Let C be a composition algebra over R with standard involution whose norm n_{C} is a Pfister form. Let m = 2^{t}. For every ∑_{i=1}^{m} \overline{z}_{i}z_{i} ∈ R^{×}, ∑_{i=1}^{m} \overline{w}_{i}w_{i} ∈ R there exist u_{1}, …, u_{m} ∈ C such that

(∑_{i=1}^{m} \overline{z}_{i}z_{i})(∑_{i=1}^{m} \overline{w}_{i}w_{i})= ∑_{i=1}^{m} \overline{u}_{i}u_{i}.

Moreover, there are v_{1}, …, v_{m} ∈ C such that

(∑_{i=1}^{m} \overline{z}_{i}z_{i})(∑_{i=1}^{m} \overline{v}_{i}v_{i}) = ∑_{i=1}^{m} \overline{w}_{i}w_{i}.

Corollary 3.9 (cf. [7, 1.9]). Let m = 2^{s} be a power of two. Let A be a generalized Cayley-Dickson algebra over R of rank 2^{s} in the sense of [8, (6.1), (6.2)] obtained from R1 by a repeated application of the (classical) Cayley-Dickson doubling process with scalar involution ∗.

(1) Let R be a semilocal ring where 2 is invertible. Then the product of two sums of m hermitian squares in (A, ∗), where one is an invertible element in R, is again a sum of m hermitian squares. In particular: Let (C, ) be a composition algebra over R. Then the product of two sums of m hermitian squares in (C, ), where one is an invertible element in R, is again a sum of m hermitian squares.

(2) Let k be a field of char(k) ≠ 2. Then the product of two sums of m hermitian squares in (A, ∗) is again a sum of m hermitian squares. In particular: Let (C, ) be a composition algebra over k. Then the product of two sums of m hermitian squares in (C, ) is again a sum of m hermitian squares.

Proof. (1) For m = 2^{s}, the form m × n_{A} with n_{A}(x)1_{A} = x^{∗}x (and in particular, m × n_{C}) is a Pfister form, and thus round [1, IV.(2.4), p.95], meaning that the set D(m × n_{A}) of invertible elements represented by the form is a group under multiplication.

(2) is now obvious.

Note that we can reduce the number of squares needed in the above proposition by one in certain cases (cf. [7, (1.10)]), since the following holds:

Proposition 3.10 ([1, p. 104]). Let R be a semilocal ring where 2 is invertible, and let C be a composition algebra over R with standard involution . Let z = (z_{1}, …, z_{m}) be such that ∑_{i=1}^{m} \overline{z}_{i}z_{i} ∈ R^{×}. Then

(∑_{i=1}^{m} \overline{z}_{i}z_{i})(∑_{i=1}^{m} \overline{w}_{i}w_{i}) = ∑_{i=1}^{m} \overline{u}_{i}u_{i}

with u_{1} = ∑_{i=1}^{m} \overline{z}_{i}w_{i} + ∑_{i=1}^{m} \overline{w}_{i}z_{i} = ⟨⟨1, …, 1⟩⟩ ⊗ n_{C}((z_{1}, …, z_{m}), (w_{1}, …, w_{m})).

Note that a form of type n_{C} ⊗⟨⟨1, …, 1⟩⟩ is not round over every ring (cf. [2]), hence the above proofs do not always work. What remains true over an arbitrary ring is the following property (which, dropping the assumption on (z_{1}, …, z_{m}) is trivially satisfied even for any integer m):

Let C be a composition algebra with standard involution over a ring R. Then the form n_{C} ⊗ ⟨⟨a_{1}, …, a_{m}⟩⟩ is the norm n_{A} of the algebra

A = Cay(C, −a_{1}, …, −a_{m})

and thus satisfies n_{A}(x^{2}) = n_{A}(x)^{2} for all x ∈ A.

In particular: The square of a sum of s = 2^{r} hermitian squares of a composition algebra C over R with standard involution is again a sum of s = 2^{r} hermitian squares, i.e. (\overline{x}_{1}x_{1} + ⋯ + \overline{x}_{s}x_{s})^{2} = \overline{z}_{1}z_{1} + ⋯ + \overline{z}_{s}z_{s}, where the vector (z_{1}, …, z_{s}) is bilinearly dependent on the vector (x_{1}, …, x_{s}).

The square of a sum of s = 2^{r} squares over any R is again a sum of s = 2^{r} squares, i.e. (x_{1}^{2} + ⋯ + x_{s}^{2})^{2} = z_{1}^{2} + ⋯ + z_{s}^{2}, where the vector (z_{1}, …, z_{s}) is bilinearly dependent on the vector (x_{1}, …, x_{s}).

[7, (1.8)] can be easily generalized to octonion division algebras over fields:

Proposition 3.11. Let C = Cay(k, a, b, c) be an octonion algebra over a field k with standard involution . Let s(k) ≥ 8. If −a, −b and −c are a sum of at most 8^{−1}s(k) squares in k then S_{h}(C, ) = 8^{−1}s(k).

Even more generally, we obtain

Proposition 3.12. Let R be a semilocal ring where 2 is invertible.

(1) Let C = Cay(R, a) be a quadratic étale algebra over R with standard involution . Let s(R) ≥ 2. If −a is a sum of at most 2^{−1}s(R) squares in R, then S_{h}(C, ) ≤ 2^{−1}s(R).

(2) Let C = Cay(R, a, b) be a quaternion algebra over R with standard involution . Let s(R) ≥ 4. If −a and −b are a sum of at most 4^{−1}s(R) squares in R, then S_{h}(C, ) ≤ 4^{−1}s(R).

(3) Let C = Cay(R, a, b, c) be an octonion algebra over R with standard involution . Let s(R) ≥ 8. If −a, −b and −c are a sum of at most 8^{−1}s(R) squares in R, then S_{h}(C, ) ≤ 8^{−1}s(R).

(4) Let C = Cay(R, a_{1}, …, a_{r}), r ≥ 2 be a generalized Cayley-Dickson algebra over R with scalar involution ∗. Let s(R) ≥ 2^{r}. If −a_{1}, …, −a_{r} are sums of at most 2^{−r}s(R) squares in R, then S_{h}(C, ) ≤ 2^{−r}s(R).

Proof. The proof is analogous to the one given in [7]. We demonstrate how to do (1) for the convenience of the reader: Since s(R) = 2m, we can write −1 = ∑_{i=1}^{m} r_{i}^{2} + ∑_{i=1}^{m} s_{i}^{2} with r_{i}, s_{i} ∈ R. Let −a = ∑_{i=1}^{m} t_{i}^{2} with t_{i} ∈ R. Thus, in case ∑_{i=1}^{m} s_{i}^{2} is invertible in R there are v_{1}, …, v_{m} ∈ C such that (∑_{i=1}^{m} t_{i}^{2})(∑_{i=1}^{m} v_{i}^{2}) = ∑_{i=1}^{m} s_{i}^{2} by Proposition 3.8, and we get

∑_{i=1}^{m} r_{i}^{2} + (∑_{i=1}^{m} t_{i}^{2})(∑_{i=1}^{m} v_{i}^{2}) = ∑_{i=1}^{m}(r_{i}^{2} − av_{i}^{2}) = ∑_{i=1}^{m} n_{C}(z_{i})

where z_{i} = (r_{i}, v_{i}). It follows that S_{h}(C, ) ≤ m.

There is another way to generalize the idea of the proof of [7, (1.8)], a similar result is mentioned in [14, (2.1)] for the quaternion case:

Proposition 3.13. Let k be a field of char(k) ≠ 2, and C = Cay(D, c) a composition division algebra, i.e. c ∈ k^{×} and D a quadratic separable field extension or a quaternion algebra. Let be the standard involution of both D and C. Assume that S_{h}(D, ) ≥ 2 and is finite.

(1) If −c is a sum of at most 2^{−1}S_{h}(D, ) hermitian squares in (D, ), then

S_{h}(C, ) = 2^{−1}S_{h}(D, ).

(2) If c is not a sum of less than or equal to S_{h}(D, ) hermitian squares in

(D, ), then

S_{h}(C, ) = S_{h}(D, ).

Also this result can easily be generalized to Cayley-Dickson algebras.

4. The Hat-involution

The hat-involution for quaternion algebras over fields of characteristic not two, used by Lewis [7], can be generalized to our more general setting as follows:

Theorem 4.1. Let R be an arbitrary ring. Let C be a composition algebra over R of rank greater than 2 which is the Cayley-Dickson doubling of a composition algebra D of half its rank, i.e. C = Cay(D, P, N) as described in 2.2. Let be the standard involution of D. Then

{ \atop (u, w)} := (\overline{u}, w)

is an involution on C, which is not scalar.

The proof is a straightforward computation which uses in particular that the multiplication on C is of the form

(u, w)(u′, w′) = (uu′ + h(w′, w), w′u + w\overline{u′})

for u, u′ ∈ D and w, w′ ∈ P, and where h : P × P → D is the -hermitian form induced by the norm N : P → R (written as unique bilinear map P × P → D which satisfies the identity (wu)(wv) = N(w)\overline{v}u in [9, (2.5)]). We call this involution the hat-involution. Note that by the definition of the hat-involution,

S_{h}(C, ) ≤ S_{h}(D, ).

A straightforward computation shows that Sym(C, ) is a projective R-module of (constant) rank 3 and Skew(C, ) a projective R-module of (constant) rank 1 if C is a quaternion algebra. If C is an octonion algebra, then Sym(C, ) is projective of (constant) rank 5, and Skew(C, ) of (constant) rank 3.

Theorem 4.2 ([11]). Let R be a ring with 2 ∈ R^{×}, and C a composition algebra of rank r over R. There is a one-one correspondence between non-standard involutions of the first kind on C and composition subalgebras of rank r/2. More precisely:

(1) Let τ be an involution on C of the first kind which is not the standard

involution . Then

C ≅ Cay(B, P, N)

with B = \{x ∈ C | τ(x) = \overline{x}\} a composition subalgebra of C, and with P = \{x ∈ C | τ(x) = −\overline{x}\}.

(2) If B is a composition subalgebra of C of rank r/2, then B determines a nontrivial automorphism f ∈ Aut(C) satisfying f^{2} = id and thus an involution τ = f ∘ of the first kind.

This implies that there exist composition algebras whose only involution of the first kind is the standard one. This is because not every quaternion or octonion algebra over a ring contains a composition subalgebra of half its rank. For examples, the reader is referred to [5], where octonion algebras over polynomial rings are constructed, which indeed only have a composition subalgebra of rank 1. See also [10].

Proposition 4.3 ([11]). Let R be a ring with 2 ∈ R^{×}, and let C be a composition algebra over R with non-standard involution τ of the first kind. Then C ≅ Cay(B, P, N) and τ((u, v)) = (\overline{u}, v) is the hat-involution on Cay(B, P, N).

As an immediate consequence it follows that in order to compute the hermitian level of a composition algebra C with arbitrary non-standard involution, it suffices to find the values of S_{h}(C, ).

Lemma 4.4. Let R be a ring with 2 ∈ R^{×}, and C a split composition algebra over R, equipped with the hat-involution. If C = Cay(D, P, N) with D split, or if C = Cay(D, μ) with μ ∈ R^{×}^{2}, then S_{h}(C, ) = 1.

Proof. If C = Cay(D, P, N) with D a split composition algebra over R, then

S_{h}(C, ) ≤ S_{h}(D, ) = 1.

If C = Cay(D, μ) for some μ ∈ R^{×}^{2} with D not necessarily split, then there exists an element v ∈ C such that \overline{v}v = −1, and such that { \atop (0, v)}(0, v) = (\overline{v}v, 0) = (−1, 0) ∈ D ⊕ D = C.

The statement of [7, (2.2)] also holds if C is an octonion algebra:

Lemma 4.5. Let k be a field of char(k) ≠ 2. If C = Cay(k, a, b, c) is a composition division algebra over k, and Cay(k, a, b, −c) is not a division algebra, then

S_{h}(C, ) = 1.

Proof. Put D = (a, b)_{k}. Then D is a division algebra, and Cay(k, a, b, −c) is no division algebra if and only if there is an element z ∈ D^{×} such that −c = \overline{z}z. View z as an element in C, and let 1, i, j, k, e be a standard basis of C, then ẑz =

\overline{z}z = −c. We compute { \atop (ze^{−1})}(ze^{−1}) = c^{−2}(e\overline{z})(ze) = c^{−2}(ze)(ze) = c^{−2}c\overline{z}z = −1

using the well-known identities e\overline{z} = ze and (ue)(ve) = c\overline{v}u for u, v ∈ D.

In particular, for any field where there is only one octonion division algebra C, it follows that S_{h}(C, ) = 1 (cf. [7, (2.3)]).

Example. Let C = Cay(ℝ, −1, −1, −1) be the real octonions. Then S_{h}(C, ) = 1.

Proposition 4.6. Let C = Cay(D, P, N) be a composition algebra over a ring R where 2 is an invertible element. Define C̃ = Cay(D, P, −N).

(1) S_{h}(C̃, ) ≤ S_{h}(C, ).

(2) If m = S_{h}(C̃, ) and if there are elements x_{i} = (u_{i}, w_{i}) ∈ C̃ such that both −1 = ∑_{i=1}^{m} \overline{x}_{i}x_{i} and 0 = ∑_{i=1}^{m} w_{i}\overline{u}_{i} for x_{i} = (u_{i}, w_{i}) ∈ C̃, then S_{h}(C, ) ≤ S_{h}(C̃, ) as well, and in particular, S_{h}(C, ) = S_{h}(C̃, ).

Proof. For an element x ∈ C = Cay(D, P, N) with x = (u, w) we compute

x̂x = (\overline{u}, w)(u, w) = (\overline{u}u + h(w, w), w\overline{u} + w\overline{u}) = (n_{D}(u) + N(w), 2w\overline{u}).

Thus we obtain for x_{i} = (u_{i}, w_{i}) ∈ C that

−1 = ∑_{i=1}^{m} x̂_{i}x_{i} = ∑_{i=1}^{m} { \atop (u_{i}, w_{i})}(u_{i}, w_{i}) = ∑_{i=1}^{m}(n_{D}(u_{i}) + N(w_{i}), 2w_{i}\overline{u}_{i})

which is equivalent to

−1 = ∑_{i=1}^{m}(n_{D}(u_{i}) + N(w_{i})) and 0 = ∑_{i=1}^{m} w_{i}\overline{u}_{i}.

The first equality, however, is equivalent to

−1 = ∑_{i=1}^{m} nC̃((u_{i}, w_{i})) = ∑_{i=1}^{m} \overline{(u_{i}, w_{i})}(u_{i}, w_{i}),

where is the standard involution on C̃.

This generalizes an observation by Serhir [14, Sec. 4.2].

Proposition 4.7 ([7, (2.4)]). For any power of two, n = 2^{ℓ}, there exists an octonion division algebra C over a field such that S_{h}(C, ) = 2^{ℓ−1}.

Proof. The proof is analogous to the proof of [7, (2.4)]. Take

k = ℝ(x_{1}, …, x_{n})(\sqrt{− ∑_{i=1}^{n} x_{i}^{2}})

with n = 2^{ℓ}. Then s(k) = n. Consider the rational function field K = k(t_{1}, t_{2}) in two variables over k and the octonion division algebra C = Cay(K, −1, −t_{1}, −t_{2}). Obviously, m := S_{h}(C, ) ≤ S_{h}(k(\sqrt{−1}), ) = 2^{ℓ−1}. A straightforward computation shows that 0 = ∑_{i=1}^{m+1} x̂_{i}x_{i} implies

0 = ∑_{i=1}^{m+1} p_{i}^{2} + q_{i}^{2} + t_{1}(r_{i}^{2} + s_{i}^{2}) + t_{2}(α_{i}^{2} + β_{i}^{2} + t_{1}(γ_{i}^{2} + δ_{i}^{2}))

for x_{i} = p_{i}+q_{i}i+r_{i}j+s_{i}k+α_{i}e+β_{i}ie+γ_{i}je+δ_{i}ke with p_{i}, q_{i}, r_{i}, s_{i}, α_{i}, β_{i}, γ_{i}, δ_{i} ∈ K. We may assume without loss of generality that these are all elements in k[t_{1}, t_{2}]. Put t_{1} = t_{2} = 0, then we obtain a sum of (2m+2) squares in k. Thus s(k) ≤ 2m+1 so that 2^{ℓ} ≤ 2m + 1, i.e. m ≥ 2^{ℓ−1} − 1/2. Since m is an integer we must have m = 2^{ℓ−1}.

It is an interesting question which values indeed are possible for the hermitian level S_{h}(C, ) of an octonion algebra C. For a quaternion algebra D over a field of characteristic not 2, it is known that S_{h}(D, ) is always a power of two, provided that it is finite [14, Corollaire 3.2].

Lemma 4.8. Let k be a field of characteristic not two such that s(k) = 2^{t} ≥ 8. Let C = Cay(k, a, b, c) be an octonion division algebra.

(1) If −a, −b and c are each a sum of at most 8^{−1}s(k) squares in k, then

2^{t−3} ≤ S_{h}(C, ) ≤ 2^{t−2}.

(2) If −c is not a sum of less than or equal to S_{h}(D, ) hermitian squares in (D, ), and if C̃ = Cay(D, −c) is a division algebra then S_{h}(C, ) is a power of 2.

Proof. (1) For C̃ = Cay(k, a, b, −c) we have S_{h}(C̃, ) = 2^{t−3} = 2^{−1}S_{h}(D, ) with D = (a, b)_{k}. Therefore the assertion follows from the fact that S_{h}(C̃, ) ≤ S_{h}(C, ) ≤ S_{h}(D, ).

(2) In this case, S_{h}(C̃, ) = S_{h}(D, ) by Proposition 3.13 applied to the algebra C̃.

Remark 4.9. Let C = Cay(k, a, b, c) be an octonion division algebra over a field. For x_{i} = p_{i}+q_{i}i+r_{i}j+s_{i}k+α_{i}e+β_{i}ie+γ_{i}je+δ_{i}ke with p_{i}, q_{i}, r_{i}, s_{i}, α_{i}, β_{i}, γ_{i}, δ_{i} ∈ k we calculate that 0 = ∑_{i=1}^{m} x̂_{i}x_{i} is equivalent to

(1)  0 = ∑_{i=1}^{m}(p_{i}^{2} − aq_{i}^{2} − br_{i}^{2} + abs_{i}^{2}) − c(α_{i}^{2} − aβ_{i}^{2} − bγ_{i}^{2} + abδ_{i}^{2})

and

(2)  0 = ∑_{i=1}^{m}(α_{i} + β_{i}i + γ_{i}j + δ_{i}k)(p_{i} − q_{i}i − r_{i}j − s_{i}k).

Now (2) is equivalent to the equations

0 = ∑_{i=1}^{m}(α_{i}p_{i} − β_{i}q_{i}α − γ_{i}r_{i}b + abδ_{i}s_{i})

0 = ∑_{i=1}^{m}(−α_{i}q_{i} + β_{i}p_{i} + γ_{i}s_{i}b − δ_{i}r_{i}b)

0 = ∑_{i=1}^{m}(α_{i}r_{i} − β_{i}s_{i}a + γ_{i}p_{i} + δ_{i}q_{i}a)

0 = ∑_{i=1}^{m}(−α_{i}s_{i} − β_{i}r_{i} + γ_{i}q_{i} + δ_{i}p_{i}).

Thus 0 is a nontrivial sum of m hermitian squares in (C, ) if and only if in k there exists a nontrivial common zero of 5 quadratic forms in 8m variables

p_{1}, …, p_{m}, q_{1}, …, q_{m}, s_{1}, …, s_{m}, r_{1}, …, r_{m},
α_{1}, …, α_{m}, β_{1}, …, β_{m}, γ_{1}, …, γ_{m}, δ_{1}, …, δ_{m}.

This indicates that it may be quite difficult to determine which values can occur as S_{h}(C, ).

[7, 4.1, 4.2] generalize to octonion algebras as follows:

Lemma 4.10. Let k be a formally real field and C = Cay(k, a, b, c) an octonion algebra over k.

(1) Let S_{h}(C, ) < ∞. (a) If −a and −b are sums of squares in k then so is c.

(b) If −c and −a (or −b) are sums of squares in k then so is b (or a).

(2) Let S_{h}(C, ) < ∞. (a) If −a and −b are sums of squares in k then so is −c.

(b) If c and a (or b) are sums of squares in k then so is −b (or −a).

Proof. (1) The proof is analogous to the one of [7, 4.1].

(2) If S_{h}(C, ) < ∞ then S(C̃, ) < ∞ for C̃ := Cay(k, a, b, −c). Hence (1) implies the assertion.

Lemma 4.11 ((cf. [7, 4.3])). Let k be a formally real field, and C = Cay(k, a, b, c) an octonion algebra over k. If both −a and −b are sums of squares in k then either S_{h}(C, ) or S_{h}(C, ) is infinite.

The proof is again analogous to the one given in [7].

Conjecture. Let k be a field of characteristic not two and let C be any octonion division algebra over k. If S_{h}(C, ) is finite, it must be a power of two.

5. Involutions of the Second Kind

We now briefly turn to involutions of the second kind on composition algebras. Involutions of the second kind on generalized Cayley-Dickson algebras and composition algebras behave special in the following way. (For quaternion algebras, this result is due to Albert, cf. [6, I.2.(2.22)].)

Theorem 5.1 ([11]). Suppose 2 is invertible in R. Let S be a quadratic étale faithfully flat R-algebra with standard involution s_{0}. Let C be a composition algebra over S of rank ≥ 4, and A = Cay(C, μ_{1}, …, μ_{m}) a generalized Cayley-Dickson algebra of rank A = 2^{m} rank C ≥ 16 (i.e. A is a noncommutative Jordan algebra). Let τ be any involution on A of the second kind.

(1) There exists a unique flexible quadratic R-subalgebra A_{0} of A with a unique scalar involution σ_{0} such that

A = A_{0} ⊗_{R} S and τ = σ_{0} ⊗ s_{0}.

The algebra A_{0} is uniquely determined by these conditions.

(2) There exists a unique composition R-subalgebra C_{0} of C such that

A = C_{0} ⊗_{R} S and τ = ⊗ s_{0},

where denotes the standard involution on C_{0}. The algebra C_{0} is uniquely determined by these conditions.

We restrict ourselves to the study of these involutions on composition algebras, since it is immediately obvious how to obtain an analogous statement also for generalized Cayley-Dickson algebras. In the above situation, let τ be an involution of the second kind on a composition algebra C over a quadratic étale R-algebra S. We know that there is a unique composition subalgebra C_{0} over R such that C ≅ C_{0} ⊗_{R} S and τ = ⊗ s_{0}. Obviously,

S_{h}(C, τ) ≤ min{S_{h}(C_{0}, ), S_{h}(S, s_{0})}.

Moreover, if the quadratic étale algebras S = Cay(R, c) is a classical Cayley-Dickson doubling of R, write x = (z, w) ∈ C = C_{0} ⊕ C_{0} (as an S-module) with z, w ∈ C_{0}. We compute τ(x) = (\overline{z}, −\overline{w}) and

τ(x)x = (n_{C_{0}}(z) − cn_{C_{0}}(w), −(\overline{z}w + \overline{w}z)) = (n_{C_{0}}(z) − cn_{C_{0}}(w), −n_{C_{0}}(z, w)).

Thus

−1 = ∑_{i=1}^{m} τ(x_{i})x_{i}

= (∑_{i=1}^{m}(n_{C_{0}}(z_{i}) − cn_{C_{0}}(w_{i})), − ∑_{i=1}^{m} n_{C_{0}}(z_{i}, w_{i}))

= (∑_{i=1}^{m} n_{A}((z_{i}, w_{i})), − ∑_{i=1}^{m} n_{C_{0}}(z_{i}, w_{i}))

with A := Cay(C_{0}, c) the generalized Cayley-Dickson algebra over R, and (z_{i}, w_{i}) ∈ A.

Lemma 5.2. In the above situation,

S_{h}(A, ∗) ≤ S_{h}(C, τ) ≤ S_{h}(C_{0}, ).

If S/R = k/k_{0} is a separable quadratic field extension, then

d_{1} ≤ 2S_{h}(C, τ) ≤ d_{2},

where d_{1} is the order of the quadratic form n_{A} in W (k), and d_{2} the order of the quadratic form n_{C_{0}} in W (k).

In particular, if c is not a sum of less than or equal to S_{h}(C_{0}, ) hermitian squares in (C_{0}, ) then S_{h}(C, τ) is a power of two.

Proof. We know that S_{h}(C_{0}, ) = S_{h}(A, ∗) by Proposition 3.13 (or, more precisely, its generalization for Cayley-Dickson algebras). Thus the assertion follows from S_{h}(A, ∗) ≤ S_{h}(C, τ) ≤ S_{h}(C_{0}, ) (Proposition 3.5).

Acknowledgements

The second author thanks the first author and the Fakultät für Mathematik, Universität Regensburg, for hospitality while this research was undertaken. Part of this research is funded by the TMR research network (ERB FMRX CT-97-0107) on “K-theory and algebraic groups”.

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S. Pumplün, Fakultät für Mathematik, Universität Regensburg, Universitätsstr. 31, D-93040 Regensburg, Germany.

E-mail address: susanne.pumpluen@mathematik.uni-regensburg.de

T. Unger, Department of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland.

E-mail address: thomas.unger@ucd.ie