INVOLUTIONS ON COMPOSITION ALGEBRAS

S. PUMPLÜN

Abstract. Involutions on composition algebras over rings where 2 is invertible are investigated. It is proved that there is a one-one correspondence between non-standard involutions of the first kind, and composition subalgebras of half rank. Every non-standard involution of the first kind is isomorphic to the natural generalization of Lewis’s hat-involution [L]. Any involution of the second kind on a composition algebra C over a quadratic etale R-algebra S can be written as the tensor product of the standard involution of a unique R-composition subalgebra of C and the standard involution of S/R. The latter generalizes a well-known theorem of Albert on quaternion algebras with unitary involutions.

Introduction

It is common knowledge that octonion algebras possess a unique scalar involution which is usually called the standard (or canonical) involution. However, there seems to be nothing explicit in the literature about involutions on octonion algebras in general. The aim of this paper is to study these involutions.

We define involutions of the first and second kind on composition algebras. For rings where 2 is an invertible element, it turns out that non-standard involutions of the first kind are in one-one correspondence with the composition subalgebras of half rank. This observation, which implies that there exist composition algebras over rings which do not have any involution of the first kind other than the standard one, is a direct consequence of the fact that the reflections on composition algebras (i.e. the automorphisms of order two) are in one-one correspondence with the composition subalgebras of half the rank of the

1991 Mathematics Subject Classification. Primary: 17A75; Secondary: 16W10.

Key words and phrases. Composition algebras, involutions, reflections.

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algebra. This is proved by Jacobson [J] in the special case where the base ring is a field. We conclude that every non-standard involution of the first kind on a composition algebra of rank 4 or 8 is isomorphic to the hat-involution.

Albert [A] proved that involutions of the second kind on a quaternion division algebra over a field are of a very special type: Let C be a quaternion division algebra with center a separable quadratic field extension F/K. If C has an involution of the second kind τ, then C = C_{0} ⊗_{K} F with C_{0} a quaternion algebra over K stabilized by τ. The involution τ is the tensor product of the standard involutions of C_{0} and F/K.

A close look at the proof of this result as given in [KMRT, (2.22)] shows that the most relevant feature of a quaternion algebra used is the fact that it possesses a unique scalar involution. Thus the result can easily be generalized to arbitrary (unital) algebras over a quadratic etale R-algebra S, which are finitely generated projective as S-modules, and have such a unique scalar involution σ. We thus obtain an analogous result for involutions of the second kind on any such algebra, and in particular, on generalized Cayley-Dickson algebras, as well as composition algebras.

The investigation of the hermitian level of composition algebras was one of the motivations of this paper (see [PU]).

The author would like to thank H.P. Petersson for his comments and suggestions.

1. Preliminaries

Let R be a unital commutative associative ring. The term ”Ralgebra” always refers to unital nonassociative algebras which are finitely generated projective as R-modules. Assume that a1_{A} = 0 implies a = 0 for every a ∈ R (cf. [M]).

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, where n(x, y) denotes the induced symmetric bilinear form n(x, y): = n(x + y) − n(x) − n(y). The form n is uniquely determined and called the norm of the quadratic algebra A.

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A map τ : A → A is called an involution on A if it is an antiautomorphism of period 2, i.e. if τ(x + y) = τ(x) + τ(y), τ(xy) = τ(y)τ(x), and τ^{2} = id, for all x, y ∈ A. It 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. In that case, n_{A}(x) = τ(x)x resp. t_{A}(x) = τ(x) + x is a quadratic, resp. a linear form on A [M, p.86]. An involution τ on A is called of the first kind if τ|_{R} = id.

Let S be a quadratic etale 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.

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) 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 non-associative algebra is called alternative if its associator [x, y, z] = (xy)z − x(yz) 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 [M, 4.6]. Composition algebras only exist in ranks 1, 2, 4 or 8. Those of rank 2 are exactly the quadratic etale R-algebras, those of rank 4 exactly the well-known quaternion algebras. The ones of rank 8 are also called octonion algebras.

A composition algebra C has 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, and x + \overline{x} = t(x)1 ∈ R1.

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

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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 new 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 starting from a composition algebra C over R we obtain either again a composition algebra (if the rank of the new algebra is at most 8), or a generalized Cayley-Dickson algebra of rank 2^{m}rank(C) ≥ 16. The latter are no longer alternative, but still flexible with a scalar involution [M].

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 [P]:

Let D be a composition algebra of rank ≤ 4 over R 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 (i.e., a biadditive map h: P ×P → D with h(ws, w′t) = s̄h(w, w′)t and h(w, w′) = \overline{h(w′, w)} for all s, t ∈ D, w, w′ ∈ P, and where P → \overline{P}^{∨}, w ↦ h(w, ·) is an isomorphism of right D-modules). Define a norm N: P → D by N(w) = h(w, w) for w ∈ P. The R-module A = D ⊕ P becomes a new R-algebra by the multiplication

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

for u, u′ ∈ D, w, w′ ∈ P, with · denoting the right D-module structure of P.

The algebra constructed above is denoted by Cay(D, P, N)

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= Cay(D, P, h). Its norm is given by the form n((u, w)) = n_{D}(u) − N(w). D itself is standardly a (free) right D-module of rank one and “norm one” (cf. [P]). Any norm on D is similar to n_{D} (resp., any nondegenerate hermitian form h: D×D → D is similar to the standard form given by the involution, i.e. to (w, w′) ↦ \overline{w}′w). In this special case the ”classical” doubling process Cay(D, μ) = Cay(D, D, μn_{D}) with μ ∈ R is obtained.

2. Involutions of the first kind

Following Jacobson [J] a non-trivial automorphism f of a composition algebra C over a ring R is called a reflection if f^{2} = id, i.e. if it has order two. The observation in [J, p.66] can be easily adapted to our more general situation:

2.1 Theorem Let R be a ring where 2 is invertible, and C a composition algebra over R of rank r. There is a one-one correspondence between reflections on C and composition subalgebras of C of rank \frac{r}{2}:

(i) Given a reflection f ∈ Aut(C), the subalgebra B = {x ∈ C|f(x) = x} of fixed elements with respect to f is a composition subalgebra of rank \frac{r}{2}, and

C ≅ Cay(B, P, N)

with

P = {x ∈ C|f(x) = −x}.

(ii) Given a composition subalgebra B of C of rank \frac{r}{2}, let P be the orthogonal complement of B with respect to the norm of C. Then C = B ⊕ P as R-modules, and

f : Cay(B, P, N) → C,
f(u + w) = u − w

for u ∈ B, w ∈ P is a reflection on C.

Proof (i) Clearly, the subalgebra B is a quadratic alternative R- subalgebra of C with norm n_{B} = n_{C}|_{B} : B → R. In the field case it is known that n_{B} is nondegenerate [J]. This implies the nondegeneracy of n_{B} also in our setting, since we may simply pass to the localisations

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and to the residue class algebras for all P ∈ Spec(R) to prove the assertion. Hence B is a composition algebra. Moreover, B locally has rank \frac{r}{2}, since again this is true for fields of characteristic not 2 by [J]. (ii) This is the generalized Cayley-Dickson doubling process [P, 2.5]. □

2.2 Corollary Let 2 be an invertible element in R, and C a composition algebra over R of rank r, which does not contain a composition algebra of rank \frac{r}{2}. Then there are no reflections on C.

From now on assume that 2 is an invertible element in R. Let τ be an involution on A of the first kind which is not the standard involution . Then τ ∘ = ∘τ. Thus the assigments τ → τ ∘ and f → f ∘ give inverse bijections between the involutions of C of the first kind and the automorphisms of C of period 2. We obtain the following result as an immediate corollary from the above theorem:

2.3 Theorem Let R be a ring where 2 is invertible, and C a composition algebra over R of rank r. There is a one-one correspondence between the non-standard involutions on C of the first kind and the composition subalgebras of rank \frac{r}{2}:

(i) Given a non-standard involution τ on C of the first kind, then

C ≅ Cay(B, P, N)

is the Cayley-Dickson doubling of a composition algebra, with

B = {x ∈ C|τ(x) = x̄},

and

P = {x ∈ C|τ(x) = −x̄}.

(ii) Given a composition subalgebra B of C of rank \frac{r}{2}, we obtain a reflection f : Cay(B, P, N) → C and an involution τ = f ∘ of the first kind on C.

2.4 Corollary Let 2 be an invertible element in R, and C a composition algebra over R of rank r, which does not contain a composition algebra of rank \frac{r}{2}. Then there are no involutions on C of the first kind other that the standard one.

Examples of such algebras can be found in [KPS1], where octonion algebras over polynomial rings in two indeterminates are constructed, which only have the ring itself as a composition subalgebra. See [Pu] for examples on quaternion algebras.

The hat-involution is defined by Lewis [L], for quaternion algebras over fields of characteristic not two. This definition is generalized as follows.

2.5 Proposition Let R be an arbitrary ring. Let C be a composition algebra over R of rank greater that 2 which is the Cayley-Dickson doubling of a composition algebra D of half its rank, i.e. C = Cay(D, P, N). Let

be the standard involution of D. Then

(u, w)ˆ = (ū, 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 that h : P × P → D satisfies the identity h(wu, wv) = h(w, w)v̄u. We call this involution the hatinvolution. If C is a quaternion algebra, then Sym(C, ˆ ) = {x ∈ C|x̄ = x} is a projective R-module of constant rank 3, and Skew(C, ˆ ) = {x ∈ C|x̄ = −x} is a projective R-module of constant rank 1, if C is an octonion algebra then Sym(C, ˆ ) is projective of constant rank 5, and Skew(C, ˆ ) projective of rank 3.

2.6 Corollary Let R be a ring where 2 is invertible, and C a composition algebra over R with non-standard involution τ of the first kind. Then

C ≅ Cay(B, P, N)

and

τ((u, v)) = (ū, v)

is the hat-involution on C.

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3. Involutions of the second kind

Let A be an S-algebra with a scalar involution σ of the first kind, with norm n_{A} : A → S, n_{A}(x) = σ(x)x. This scalar involution is unique, since A is a quadratic S-algebra whose norm n_{A} is uniquely determined [P, 1.2]. Let τ be an involution of the second kind on A. Take an element x ∈ A. Let y = τ(x). Then x(τ ∘ σ ∘ τ(x)) = τ(τ(x))τ(σ(τ(x))) = τ(y)τ(σ(y)) = τ(σ(y)y) = s_{0}(n_{A}(y)) ∈ S · 1 with n_{A} the norm of A and hence also x + τ ∘ σ ∘ τ(x) = s_{0}(t_{A}(y)) ∈ S · 1. Therefore, τ ∘ σ ∘ τ is a scalar involution on C with τ ∘ σ ∘ τ|_{S} = id. This implies [M, (1.1)]

(1)  x^{2} − t_{1}(x)x + n_{1}(x)1 = 0 , n_{1}(1_{C}) = 1,

with t_{1}(x): = x + τ ∘ σ ∘ τ(x), n_{1}(x): = x(τ ∘ σ ∘ τ(x)), i.e., (A, n_{1}) is a quadratic algebra. Since the norm n_{A} of the quadratic algebra A is uniquely determined by condition (1), it must agree with the norm n_{1} on A. It follows that τ ∘ σ ∘ τ = σ. so τ and σ commute, and σ̃ = τσ is a descent map over R, for the Galois group Gal(S/R) = ℤ/2ℤ (viewed as group schemes over R). The R-subalgebra

A_{0} = {x ∈ A | τσ(x) = x}

of A is the descent of σ̃, and the map

ϕ : A_{0} ⊗_{R} S → A, x ⊗ a → ax

is an S-algebra isomorphism (see [KPS2] for a short description of descent, or [K]). A_{0} is a quadratic R-algebra with unique norm n_{A_{0}} = n_{A}|_{A_{0}} and unique scalar involution σ_{0} = σ|_{A_{0}}. Obviously, we have n_{A_{0}} ⊗ id_{S} = n_{A}. Since these conditions imply that every element in A_{0} is invariant under τ ∘ σ, the algebra A_{0} is uniquely determined by τ. Moreover, ^{τ}A = A.

A straightforward computation shows that, indeed, τ = σ_{0} ⊗ s_{0}. We thus proved that involutions of the second kind on algebras with a scalar involution behave in a special way as follows:

3.1 Proposition Let S be a quadratic etale R-algebra, and A a (quadratic) S-algebra with a scalar involution σ of the first kind. Let τ be any involution on A of the second kind. Then there exists a unique

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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.

Note that the following converse of this result is obvious: If A = A_{0} ⊗_{R} S, where A_{0} is a quadratic R-algebra with a scalar involution σ_{0} of the first kind, then there is exactly one involution τ on A of the second type such that τ|_{A_{0}} = σ_{0} is the scalar involution.

Moreover, the proof of the above proposition only requires the existence of an involution of the first and one of the second kind which commute.

As an immediate application of the above result it turns out that involutions of the second kind on composition algebras, and more generally, on generalized Cayley-Dickson algebras, are well behaved.

3.2 Theorem Suppose 2 is invertible in R. Let S be a quadratic etale R-algebra, C 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 (so A is a noncommutative Jordan algebra). Let σ = denote the scalar involution σ on A. Let τ be any involution on A of the second kind.

(i) 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. (ii) Let S/R be faithfully flat. There exists a unique composition Rsubalgebra C_{0} of C such that

C = C_{0} ⊗_{R} S and τ = σ_{0} ⊗ s_{0}.

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

This generalizes a well-known theorem by Albert [KMRT, I.2.(2.22)] for quaternion algebras over fields.

Proof (i) is obvious, since the algebra A itself is flexible by [M, 6.8(iv)].

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(ii) C_{0} certainly is a quadratic alternative algebra with multiplicative norm n_{C_{0}}. The norm n_{C_{0}} satisfies n_{C_{0}}⊗_{R}id_{S} = n_{C} and is nondegenerate since S/R is faithfully flat. □

Note the following: If rank C = 4 then C_{0} must be a quaternion algebra, for any quadratic etale algebra S/R (cf. [KPS, p.79]).

References

[A] Albert, A.A.,“Structure of algebras”, AMS Colloq. Publ. 24, Providence, RI.

[J] Jacobson, N., Composition algebras and their automorphisms, Rend. Circ. Mat. Palermo 7, 55–80 (1958).

[K] Knus, M.A., Ojanguren, M., Theorie de la descente et Algebres d’Azumaya, Lec. Notes in Math. 389, Springer-Verlag, Berlin-Heidelberg-New York.

[KPS1] Knus, M., Parimala, R., Srinivas, V., On compositions and triality, J. Reine Angew. Math. 457, 45–70 (1994).

[KPS2] Knus, M, Parimala, R., Srinivas, V., Azumaya Algebras with Involutions, J. Algebra 130, 65–82 (1990).

[L] Lewis, D., Sums of hermitian squares, J. Algebra 115, 466–480 (1988).

[M] McCrimmon, K., Nonassociative algebras with scalar involution, Pacific J. of Math. 116 (1), 85–108 (1985).

[MPP] Morandi, P.J., Perez-Izquierdo, J.M., Pumpluen, S., On the tensor product of composition algebras, J. Algebra 243, 41–68 (2001).

[KMRT] Knus, M.A., Merkurjev, A., Rost, M., Tignol, J.-P., “The Book of Involutions”, AMS Coll. Publications, Vol.44 (1998).

[P] Petersson, H., Composition algebras over algebraic curves of genus 0, Trans. Amer. Math. Soc. 337, 473–491 (1993).

[Pu] Pumpluen, S., Quaternion algebras over elliptic curves, Comm. Algebra 26 (12), 4357–4373 (1998).

[PU] Pumpluen, S., Unger, T., The hermitian level of composition algebras, submitted.

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany, susanne.pumpluen@mathematik.uni-regensburg.de