Involutions and Trace Forms on Exterior Powers

of a Central Simple Algebra

R. Skip Garibaldi, Anne Quéguiner-Mathieu, Jean-Pierre Tignol

Received: February 2, 2001 Revised: April 25, 2001

Communicated by Ulf Rehmann

Abstract. For A a central simple algebra of degree 2n, the nth exterior power algebra λ^{n}A is endowed with an involution which provides an interesting invariant of A. In the case where A is isomorphic to Q ⊗ B for some quaternion algebra Q, we describe this involution quite explicitly in terms of the norm form for Q and the corresponding involution for B.

1991 Mathematics Subject Classification: 16K20 (11E81 20G05) Keywords and Phrases: Trace forms, involutions, central simple algebras

The classification of irreducible representations of a split semisimple simply connected algebraic group G over an arbitrary field F is well-known: they are in one-to-one correspondence with the cone of dominant weights of G. Furthermore, one can tell whether or not an irreducible representation is orthogonal or symplectic (= supports a G-invariant bilinear form which is respectively symmetric or skew-symmetric) by inspecting the corresponding dominant weight [11, §3.11]. (Throughout this paper, we only consider fields of characteristic ≠ 2, cf. 1.8.) A G-invariant bilinear form on an irreducible representation is necessarily unique up to a scalar multiple. If the assumption that G is split is dropped, then the Galois group Γ of a separable closure F_{s} of F over F acts on the cone of dominant weights (via the so-called “∗-action”), and this action may be nontrivial. Those irreducible representations corresponding to dominant weights which are not fixed by Γ are not defined over F. Although an irreducible representation ρ whose dominant weight is fixed by Γ may not be F -defined, there is always some central simple F -algebra A and a map G → SL_{1}(A) defined over F which is an appropriate descent of ρ, see [14] or [12, p. 230, Prop. 1] for details. The algebra A is uniquely determined up to F -isomorphism. If ρ is orthogonal or symplectic over F_{s}, then it is easy to show that A supports a unique G-invariant involution γ of the first kind which is adjoint to the G-invariant bilinear form over every extension of F where A is split and hence ρ is defined. It is of interest to determine γ. For example, invariants of γ in turn provide invariants of G. All involutions γ have been implicitly determined for F = ℚ_{p} and F = ℝ in [4] and [5], but over an arbitrary field the problem is much more difficult since involutions are no longer classified by their classical invariants [2]. We restrict our attention to simply connected groups of type ^{1}A_{2n−1}; that is, to the case G = SL_{1}(A) for A a central simple F -algebra of degree 2n. Moreover, we will focus on the fundamental irreducible representation corresponding to the middle vertex of the Dynkin diagram of G, which supports a G-invariant involution γ. For any nonnegative integer k ≤ 2n, there is a central simple F -algebra λ^{k}A attached to A called the kth exterior power of A, and the appropriate analogues of the fundamental representations of SL_{1}(A) are the natural maps SL_{1}(A) → SL_{1}(λ^{k}A) for 1 ≤ k < 2n. The representation we will study, which corresponds to the middle vertex of the Dynkin diagram, is the k = n case. In general, λ^{k}A is of degree \binom{2n}{k} and is Brauer-equivalent to A^{⊗k}, see [7, 10.A]. It is defined so that when A is the split algebra A = End_{F} (W), this λ^{k} End_{F} (W) is naturally isomorphic to End_{F} (∧^{k}W). The nth exterior power λ^{n}A is endowed with a canonical involution γ such that when A is split, γ is adjoint to the bilinear form θ defined on ∧^{n}W by the equation θ(x_{1} ∧ … ∧ x_{n}, y_{1} ∧ … ∧ y_{n})e = x_{1} ∧ … ∧ x_{n} ∧ y_{1} ∧ … ∧ y_{n}, where e is any basis of the 1-dimensional vector space ∧^{2n}W. This involution is preserved by the image of G in SL_{1}(λ^{n}A) and is the one we wish to describe. If n is even and A^{⊗n} is split, then γ is orthogonal and λ^{n}A is split, so our fundamental representation of G is defined over F and orthogonal. For example, for A a biquaternion algebra over an arbitrary field F, γ is adjoint to an Albert form of A [3, 6.2]. In this paper, we provide a complete description of γ for G of type ^{1}A_{2n−1} when n is odd (see 1.1) or when n is even and A is isomorphic to B ⊗ Q where Q is a quaternion algebra (in 1.4 and 1.5). In particular, until now a description of γ has not been known for any algebra A of index ≥ 8. If A is a tensor product of quaternion algebras, we provide (in 1.6 below) a formula that gives γ in terms of the norm forms of the quaternion algebras.

Describing this particular involution γ is also interesting from the point of view of groups of type ^{1}D_{2n}. Such a group is isogenous to G = Spin(E, σ) for E a central simple algebra of degree 4n and σ an orthogonal involution with trivial discriminant. If σ is hyperbolic, then E is isomorphic to M_{2}(A) for some algebra A of degree 2n. The analogue of the direct sum of the two half-spin representations for Spin(M_{2}(A), σ) over F is the map G → SL_{1}(C(M_{2}(A), σ)) where C(M_{2}(A), σ) denotes the even Clifford algebra of (M_{2}(A), σ). This algebra is endowed with a canonical involution \underline{σ} which is G-invariant; it is mostly hyperbolic but contains a nontrivial piece which is isomorphic to (λ^{n}A, γ). Please see [3] for a precise statement and [10] for a rational proof. This relationship between representations of D_{2n} and A_{2n−1} as well as the results in this paper hint at a general theory of orthogonal representations of semisimple algebraic groups over arbitrary fields. We hope to study this in the future.

1 Statement of the main results

We will always assume that our base field F has characteristic ≠ 2 and that A is a central simple F -algebra of degree 2n. (See 1.8 for a discussion of the characteristic 2 case.) We assume moreover that A is isomorphic to a tensor product A = Q⊗B, where Q is a quaternion algebra over F, and B is a central simple F -algebra, necessarily of degree n. Note that this is always the case when n is odd. We write γ_{Q} for the canonical symplectic involution on Q and n_{Q} for the norm form. If n is odd, the main result is the following, proven in Section 4:

Theorem 1.1. If n is odd, the algebra with involution (λ^{n}(Q ⊗ B), γ) is Wittequivalent to (Q, γ_{Q})^{⊗n}.

Witt-equivalence for central simple algebras is the natural generalization of Witt-equivalence for quadratic forms, see [1] for a definition. Assume now that n is even, n = 2m. Then λ^{n}A is split and the involution γ is orthogonal. We fix some quadratic form q_{A} to which γ is adjoint. It is only defined up to similarity. The algebra λ^{m}B is endowed with a canonical involution which we denote by γ_{m}. For k = 0, …, n, we let t_{k} : λ^{k}B → F be the reduced trace quadratic form defined by

t_{k}(x) = Trd_{λ^{k}B}(x^{2}). (1.2)

This form also has a natural description from the representation-theoretic viewpoint: The group SL_{1}(B) acts on the vector space λ^{k}B, and when B is split λ^{k}B is isomorphic to a tensor product of an irreducible representation with its dual, see Section 2. Consequently, there is a canonical SL_{1}(B)-invariant quadratic form on λ^{k}B; it is t_{k}. We let t_{m}^{+} and t_{m}^{−} denote the restrictions of t_{m} to the subspaces Sym(λ^{m}B, γ_{m}) and Skew(λ^{m}B, γ_{m}) of elements of λ^{m}B which are respectively symmetric and skew-symmetric under γ_{m}, so that t_{m} = t_{m}^{+} ⊕ t_{m}^{−}. The forms thus defined are related by the following equation, proven in 5.5:

Theorem 1.3. In the Witt ring of F, the following equality holds:

⟨2⟩ · ∑_{k=0}^{m−1}(−1)^{k}t_{k} = \begin{cases}−t_{m}^{−} & if m is even, \\ t_{m}^{+} & if m is odd. \end{cases}

The similarity class of q_{A} is determined by the following theorem, proven in 5.7:

Theorem 1.4. If n is even, n = 2m, the similarity class of q_{A} contains the quadratic form:

t_{m}^{+} − t_{m}^{−} + n_{Q} · (t_{m}^{−} + ∑\begin{matrix}0≤k<m \\ k even \end{matrix}⟨2⟩t_{k}) if m is even,
t_{m}^{−} − t_{m}^{+} + n_{Q} · (∑\begin{matrix}0≤k<m \\ k even \end{matrix}⟨2⟩t_{k}) if m is odd.

The Witt class of this quadratic form can be described more precisely under some additional assumptions (see Proposition 6.1 for precise statements). We just mention here a particular case in which the formula reduces to be quite nice. Assume that m is even and B is of exponent at most 2. Then λ^{m}B is split, and its canonical involution is adjoint to a quadratic form q_{B}. Even though this form is only defined up to a scalar factor, its square is actually defined up to isometry. We then have the following, proven in 5.8:

Corollary 1.5. If m is even (i.e., deg B ≡ 0 mod 4) and B is of exponent at most 2, then the similarity class of q_{A} contains a form whose Witt class is q_{B}^{2} + n_{Q} (2^{n−2} − \frac{1}{2} \binom{n}{m} − ∧^{2}q_{B}).

Some of the notation needs an explanation. For a quadratic form q on a vector space W with associated symmetric bilinear form b so that q(w) = b(w, w), we have an induced quadratic form on ∧^{2}W which we denote by ∧^{2}q. For x_{1}, x_{2}, y_{1}, y_{2} ∈ W, its associated symmetric bilinear form ∧^{2}b is defined by

(∧^{2}b)(x_{1} ∧ x_{2}, y_{1} ∧ y_{2}) = b(x_{1}, y_{1})b(x_{2}, y_{2}) − b(x_{1}, y_{2})b(x_{2}, y_{1}).

Thus if q = ⟨α_{1}, …, α_{n}⟩, we have

∧^{2}q ≃ ⊕_{1≤i<j≤n}⟨α_{i}α_{j}⟩.

From this, one sees that even if q is just defined up to similarity, ∧^{2}q is welldefined up to isometry. (The form ∧^{2}q also admits a representation-theoretic description: It is isomorphic to a scalar multiple of the Killing form on the Lie algebra 𝔬(q), where the scalar factor depends only on the dimension of q.) From Corollary 1.5, we also get the following, which is proven in 6.3:

Corollary 1.6. Let A_{r} = Q_{1} ⊗ ⋯ ⊗ Q_{r} be a tensor product of r quaternion F -algebras, where r ≥ 3, and let T_{A_{r}} be the reduced trace quadratic form on A_{r}. The similarity class of q_{A_{r}} contains a quadratic form whose Witt class is

2^{n−1} − \frac{2^{n−2}}{n} ⟨2^{r}⟩ · T_{A_{r}} = 2^{f(r)}(2^{r} − (2 − n_{Q_{1}}) ⋯ (2 − n_{Q_{r}})),

where n = 2^{r−1} = \frac{1}{2} deg A and f(r) = 2^{r−1} − r − 1.

In particular, for r = 3, we get the quadratic form

4(n_{Q_{1}} + n_{Q_{2}} + n_{Q_{3}}) − 2(n_{Q_{1}}n_{Q_{2}} + n_{Q_{1}}n_{Q_{3}} + n_{Q_{2}}n_{Q_{3}}) + n_{Q_{1}}n_{Q_{2}}n_{Q_{3}}.

Adrian Wadsworth had casually conjectured a description of q_{A_{3}} in [3, 6.8], and we now see that his conjecture was not quite correct in that it omitted the n_{Q_{1}}n_{Q_{2}}n_{Q_{3}} term.

As a consequence of Corollary 1.6, we can show that the form q_{A} lies in the nth power of the fundamental ideal of the Witt ring W F for many central simple algebras A of degree 2n; the following result is proven in 6.4:

Corollary 1.7. Suppose that A is a central simple algebra of degree 2n ≡ 0 mod 4 which is isomorphic to matrices over a tensor product of quaternion algebras. Then the form q_{A} lies in I^{n}F.

The first author conjectured [3, 6.6] that q_{A} lies in I^{n}F for all central simple F -algebras A of degree 2n ≡ 0 mod 4 and such that A^{⊗2} is split. Corollary 1.7 fails to prove the full conjecture because for every integer r ≥ 3 there exists a division algebra A of degree 2^{r} and exponent 2 such that A doesn’t decompose as A′ ⊗ A′′ for any nontrivial division algebras A′ and A′′ [6, 3.3], so such an A doesn’t satisfy the hypotheses of Corollary 1.7. If A is a tensor product of two quaternion algebras, the form q_{A} is an Albert form of A, and the Witt index of q_{A} determines the Schur index of A, as Albert has shown (see for instance [7, (16.5)]). Corollary 1.6 shows that one cannot expect nice results relating the Witt index of q_{A_{r}} and the Schur index of A_{r} for r ≥ 3. As pointed out to us by Jan van Geel, the difficulty is that Merkurjev has constructed in [9, §3] algebras of the form A_{r} for r ≥ 3 (i.e., tensor products of at least 3 quaternion algebras) which are skew fields but whose center, F, has I^{3}F = 0. By Corollary 1.7, the forms q_{A_{r}} are then hyperbolic.

Remark 1.8 (characteristic 2). One might hope that results concerning representations of algebraic groups would not involve the restriction that the characteristic is not 2. However, removing this restriction for the results in this paper would necessarily dramatically change their nature. For example, the trace forms t_{k} occurring here are degenerate in characteristic 2. Also, our methods require the ability to take tensor products of quadratic forms and to scale by a factor of ⟨2⟩, neither of which are available in characteristic 2. These restrictions may be avoidable, but we have chosen not to attempt to do so because such an attempt would almost certainly make this paper so technical that it would be nearly unreadable.

2 Description of λ^{n}M_{2}(B)

In order to prove these results, we have to describe the algebra with involution (λ^{n}(Q ⊗ B), γ), which we will do by Galois descent. Hence we first give a description of λ^{n}M_{2}(B), see Theorem 2.5 below.

Assume B = End_{F} (V) for some n-dimensional vector space V. For 0 ≤ k ≤ n, we have λ^{k}B = End_{F} (∧^{k}V). We identify M_{2}(B) ≃ End_{F} (V ⊕ V) by mapping ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(B) to the endomorphism

(x, y) ↦ (a(x) + b(y), c(x) + d(y)).

The distinguished choice of embedding of B in M_{2}(B) corresponds with the obvious choice of direct sum decomposition of V ⊕V. (There are many others.) This gives an identification λ^{n}M_{2}(B) = End_{F} (∧^{n}(V ⊕V)). For all integers k, ℓ, this decomposition determines ∧^{k}V ⊗∧^{ℓ}V as a vector subspace of ∧^{k+ℓ}(V ⊕V) by mapping (x_{1} ∧ ⋯ ∧ x_{k}) ⊗ (y_{1} ∧ ⋯ ∧ y_{ℓ}) to

(x_{1}, 0) ∧ ⋯ ∧ (x_{k}, 0) ∧ (0, y_{1}) ∧ ⋯ ∧ (0, y_{ℓ}) ∈ ∧^{k+ℓ}(V ⊕ V).

In particular, we have

∧^{n} (V ⊕ V) = ⊕_{k=0}^{n} (∧^{k}V ⊗ ∧^{n−k}V ) . (2.1)

For each k, the space ∧^{k}V ⊗∧^{n−k}V can be identified to End_{F} (∧^{k}V) as follows. Fix a nonzero element (hence a basis) e of ∧^{n}V and define a bilinear form

θ_{k}: ∧^{k} V × ∧^{n−k}V → F

by the equation

θ_{k}(x_{k}, x_{n−k}) e = x_{k} ∧ x_{n−k} for x_{ℓ} ∈ ∧^{ℓ}V.

This form is nonsingular, so it provides the identification mentioned above

∧^{k} V ⊗ ∧^{n−k}V = End_{F} (∧^{k}V) (2.2)

by sending x_{k} ⊗ x_{n−k} to the map y ↦ x_{k}θ_{n−k}(x_{n−k}, y). The product in End_{F} (∧^{k}V) then corresponds in ∧^{k}V ⊗ ∧^{n−k}V to

(x_{k} ⊗ x_{n−k})(y_{k} ⊗ y_{n−k}) = θ_{n−k}(x_{n−k}, y_{k}) x_{k} ⊗ y_{n−k}.

From (2.1) and (2.2), we deduce an identification of the corresponding endomorphism rings

λ^{n}M_{2}(B) = End_{F} (⊕_{k=0}^{n}λ^{k}B). (2.3)

This remains true in the case when B is non split, as we will prove by Galois descent. First, we must introduce some maps on ⊕_{k=0}^{n}λ^{k}B. Since the bilinear form θ_{k} is nonsingular, for any f ∈ End_{F} (∧^{k}V), we have a unique element γ_{k}(f) ∈ End_{F} (∧^{n−k}V) such that

θ_{k} (f(x), y) = θ_{k} (x, γ_{k}(f)(y)) ,

for every x ∈ ∧^{k}V and y ∈ ∧^{n−k}V. This defines a canonical anti-isomorphism (not depending on the choice of e)

γ_{k}: End_{F} (∧^{k}V) → End_{F} (∧^{n−k}V)

such that

γ_{k}(x ⊗ y) = (−1)^{k(n−k)}y ⊗ x (2.4)

for x and y as before. One may easily verify that γ_{n−k} ∘ γ_{k} = Id_{End_{F} (∧^{k}V)} for all k = 0, …, n. By Galois descent, the maps γ_{k} are defined even when B is nonsplit, i.e., we have anti-isomorphisms γ_{k}: λ^{k}B → λ^{n−k}B such that γ_{k} ∘ γ_{n−k} = Id_{λ^{k}B} (see [7, Exercise 12, p. 147] for a rational definition). In the particular case where n is even, by definition of the bilinear form θ_{n/2}, the map γ_{n/2} is actually the canonical involution on λ^{n/2}B.

Theorem 2.5. Whether or not B is split, there is a canonical isomorphism

Φ: λ^{n}M_{2}(B) → End_{F} (λ^{0}B ⊕ ⋯ ⊕ λ^{n}B)

which in the split case is the identification (2.3) above. The canonical involution γ on λ^{n}M_{2}(B) induces via Φ an involution on End_{F} (⊕_{k=0}^{n}λ^{k}B) which is adjoint to the bilinear form T defined on λ^{0}B ⊕ ⋯ ⊕ λ^{n}B by

T (u, v) = \begin{cases}(−1)^{ℓ} Trd_{λ^{k}B} (uγ_{ℓ}(v)) & if k + ℓ = n, \\ 0 & if k + ℓ ≠ n, \end{cases}

for any u ∈ λ^{k}B and v ∈ λ^{ℓ}B.

Proof. We prove this by Galois descent. Fix a separable closure F_{s} of F and let Γ := Gal(F_{s}/F) be the absolute Galois group. We fix a vector space V over F such that dim_{F} V = deg B = n and let V_{s} = V ⊗_{F} F_{s}. We fix also an F_{s}-algebra isomorphism φ: B ⊗_{F} F_{s} ⟶^{∼} End_{F} (V) ⊗_{F} F_{s}. Every σ ∈ Γ acts canonically on V_{s} and End_{F_{s}}(V_{s}) = End_{F} (V) ⊗_{F} F_{s}; we denote again by σ these canonical actions, so that σ(f) = σ ∘ f ∘ σ^{−1} for f ∈ End_{F_{s}}(V_{s}). On the other hand, the canonical action of Γ on B ⊗_{F} F_{s} corresponds under φ to some twisted action ∗ on End_{F_{s}}(V_{s}). Since every F_{s}-linear automorphism of End_{F_{s}}(V_{s}) is inner, we may find g_{σ} ∈ GL(V_{s}) such that

σ ∗ f = g_{σ} ∘ σ(f) ∘ g_{σ}^{−1} = Int(g_{σ}) ∘ σ(f) for all f ∈ End_{F_{s}}(V_{s}).

Then φ induces an F -algebra isomorphism from B onto the F -subalgebra

{f ∈ End_{F_{s}}(V_{s}) | g_{σ} ∘ σ(f) ∘ g_{σ}^{−1} = f for all σ ∈ Γ}.

The ∗-action of Γ on End_{F_{s}}(V_{s}) induces twisted actions on End_{F_{s}}(∧^{n}(V_{s} ⊕ V_{s})) and on End_{F_{s}}(⊕_{k=0}^{n} End_{F_{s}}(∧^{k}V_{s})) such that the F -algebras of Γ-invariant elements are λ^{n}(M_{2}(B)) and End_{F} (⊕_{k=0}^{n}λ^{k}B) respectively. To prove the first assertion of the theorem, we will show that these actions correspond to each other under the isomorphism

End_{F_{s}}(∧^{n}(V_{s} ⊕ V_{s})) ⟶^{∼} End_{F_{s}}(⊕_{k=0}^{n} End_{F_{s}}(∧^{k}V_{s}))

derived from (2.1) and (2.2). For σ ∈ Γ and k = 0, …, n, define ∧^{k}g_{σ} ∈ GL(∧^{k}V_{s}) by

∧^{k}g_{σ}(x_{1} ∧ … ∧ x_{k}) = g_{σ}(x_{1}) ∧ … ∧ g_{σ}(x_{k}).

Then φ induces an F -algebra isomorphism from λ^{k}B onto the F -subalgebra

{f ∈ End_{F_{s}}(∧^{k}V_{s}) | ∧^{k}g_{σ} ∘ σ(f) ∘ (∧^{k}g_{σ})^{−1} = f for all σ ∈ Γ},

hence also from End_{F} (⊕_{k=0}^{n}λ^{k}B) to

{f ∈ End_{F_{s}}(⊕_{k=0}^{n} End_{F_{s}}(∧^{k}V_{s})) |

(⊕_{k} Int(∧^{k}g_{σ})) ∘ σ(f) = f ∘ (⊕_{k} Int(∧^{k}g_{σ})) for all σ ∈ Γ}.

Similarly, define ∧^{n}(g_{σ} ⊕ g_{σ}) ∈ GL(∧^{n}(V_{s} ⊕ V_{s})) by

∧^{n} (g_{σ} ⊕ g_{σ})((x_{1}, y_{1}) ∧ … ∧ (x_{n}, y_{n})) =

(g_{σ}(x_{1}), g_{σ}(y_{1})) ∧ … ∧ (g_{σ}(x_{n}), g_{σ}(y_{n})),

so that λ^{n}(M_{2}(B)) can be identified through φ with

{f ∈ End_{F_{s}}(∧^{n}(V_{s} ⊕ V_{s})) |

∧^{n} (g_{σ} ⊕ g_{σ}) ∘ σ(f) = f ∘ ∧^{n}(g_{σ} ⊕ g_{σ}) for all σ ∈ Γ}.

Certainly, ∧^{n}(g_{σ} ⊕ g_{σ}) = ⊕_{k=0}^{n}(∧^{k}g_{σ} ⊗ ∧^{n−k}g_{σ}) under (2.1), and computation shows that ∧^{k}g_{σ} ⊗ ∧^{n−k}g_{σ} = (det g_{σ}) Int(∧^{k}g_{σ}) under (2.2). Therefore, (2.1) and (2.2) induce an isomorphism of F -algebras

Φ: λ^{n}(M_{2}(B)) ⟶^{∼} End_{F} (⊕_{k=0}^{n}λ^{k}B).

To complete the proof of the theorem, we show that the canonical involution γ on λ^{n}(M_{2}(B)) corresponds to the adjoint involution with respect to T under Φ. In order to do so, we view λ^{n}(M_{2}(B)) and End_{F} (⊕_{k=0}^{n}λ^{k}B) as the fixed subalgebras of End_{F_{s}}(∧^{n}(V_{s} ⊕ V_{s})) and End_{F_{s}}(⊕_{k=0}^{n} End_{F_{s}}(∧^{k}V_{s})), and show that the canonical involution γ on End_{F_{s}}(∧^{n}(V_{s} ⊕ V_{s})) corresponds to the adjoint involution with respect to T (extended to F_{s}) under the isomorphism induced by (2.1) and (2.2).

Taking any nonzero element e ∈ ∧^{n}V_{s}, the identification ∧^{2n}(V_{s}⊕V_{s}) = ∧^{n}V_{s}⊗ ∧^{n}V_{s} allows us to write e ⊗ e for a nonzero element of ∧^{2n}(V_{s} ⊗ V_{s}). Then γ is adjoint to the bilinear form

Θ: ∧^{n} (V_{s} ⊕ V_{s}) × ∧^{n}(V_{s} ⊕ V_{s}) → F_{s}

given by

Θ(x, y) e ⊗ e = x ∧ y for x, y ∈ ∧^{n}(V_{s} ⊕ V_{s})

as was mentioned in the introduction. Using the identification of ∧^{k}V_{s}⊗∧^{n−k}V_{s} as a subspace of ∧^{n}(V_{s} ⊕ V_{s}), we have that for x_{i}, y_{i} ∈ ∧^{i}V_{s},

Θ(x_{k} ⊗ x_{n−k}, y_{ℓ} ⊗ y_{n−ℓ}) =

\{ \begin{matrix}(−1)^{ℓ}θ_{k}(x_{k}, y_{ℓ})θ_{n−k}(x_{n−k}, y_{n−ℓ}) & if k + ℓ = n, \\ 0 & if k + ℓ ≠ n. \end{matrix}

We translate this into terms involving B, using the isomorphism φ to identify λ^{k}B_{s} := (λ^{k}B) ⊗_{F} F_{s} with End_{F_{s}}(∧^{k}V_{s}). In particular, we know that

Trd_{λ^{k}B_{s}}(x_{k} ⊗ x_{n−k}) = θ_{n−k}(x_{n−k}, x_{k})

for Trd the reduced trace, and that

θ_{k}(x_{k}, x_{n−k}) = (−1)^{k(n−k)}θ_{n−k}(x_{n−k}, x_{k}).

So for x = x_{k} ⊗ x_{n−k} ∈ λ^{k}B_{s} and y = y_{ℓ} ⊗ y_{n−ℓ} ∈ λ^{ℓ}B_{s},

Θ(x, y) = \{ \begin{matrix}(−1)^{ℓ} Trd_{λ^{k}B_{s}}(γ_{ℓ}(y)x) & if k + ℓ = n, \\ 0 & if k + ℓ ≠ n. \end{matrix}

Of course, in the k + ℓ = n case we could just as easily have taken

Θ(x, y) = (−1)^{ℓ} Trd_{λ^{ℓ}B_{s}}(γ_{k}(x)y).

So, the vector space isomorphism derived from (2.1) and (2.2) is an isometry of Θ and T, and it follows that the canonical involution γ adjoint to Θ corresponds to the adjoint involution to T under Φ.

For later use, we prove a little bit more about this isomorphism Φ. Let us consider the elements e_{1} = ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) and e_{2} = ( \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ) ∈ M_{2}(B), and let t be an indeterminate over F. We write λ^{n} for the map M_{2}(B) → λ^{n}M_{2}(B) defined in [7, 14.3], which is a homogeneous polynomial map of degree n. In the split case where M_{2}(B) is identified with End_{F} (V ⊕ V) and λ^{n}M_{2}(B) with End_{F} (∧^{n}(V ⊕ V)), the map is given by

(λ^{n}f)(w_{1} ∧ ⋯ ∧ w_{n}) = f(w_{1}) ∧ ⋯ ∧ f(w_{n})

for f ∈ End_{F} (V ⊕ V) and w_{1}, …, w_{n} ∈ V ⊕ V. Whether or not B is split, there exist ℓ_{0}, …, ℓ_{n} ∈ λ^{n}M_{2}(B) such that

λ^{n}(e_{1} + te_{2}) = t^{n}ℓ_{0} + t^{n−1}ℓ_{1} + ⋯ + tℓ_{n−1} + ℓ_{n}.

We then have Lemma 2.6. For k = 0, …, n, the image of ℓ_{k} under Φ is the projection on λ^{k}B. Moreover, we have γ(ℓ_{k}) = ℓ_{n−k}.

Proof. It is enough to prove it in the split case. Hence, we may assume B = End_{F} (V), and use identification (2.2) of the previous section. An element of λ^{k}B = End_{F} (∧^{k}V) can be written as (x_{1} ∧ ⋯ ∧ x_{k}) ⊗ (y_{1} ∧ ⋯ ∧ y_{n−k}), where x_{1}, …, x_{k}, y_{1}, …, y_{n−k} ∈ V. The endomorphism λ^{n}(e_{1} + te_{2}) acts on this element as follows:

λ^{n}(e_{1} + te_{2}) ((x_{1} ∧ ⋯ ∧ x_{k}) ⊗ (y_{1} ∧ ⋯ ∧ y_{n−k}))
= (x_{1}, 0) ∧ ⋯ ∧ (x_{k}, 0) ∧ (0, ty_{1}) ∧ ⋯ ∧ (0, ty_{n−k})
= t^{n−k}(x_{1} ∧ ⋯ ∧ x_{k}) ⊗ (y_{1} ∧ ⋯ ∧ y_{n−k}).

Hence, the image under ℓ_{i} of this element is itself if i = k and 0 otherwise. This proves the first assertion of the lemma. By Theorem 2.5, to prove the second one, one has to check that for any u, v ∈ λ^{0}B ⊕ ⋯ ⊕ λ^{n}B, we have T (ℓ_{i}(u), v) = T (u, ℓ_{n−i}(v)), which follows easily from the description of T given in that theorem.

Remark 2.7. By the previous lemma, the elements ℓ_{0}, …, ℓ_{n} ∈ λ^{n}M_{2}(B) are orthogonal idempotents. Hence, the fact that γ(ℓ_{k}) = ℓ_{n−k} for all k = 0, …, n implies that the involution γ is hyperbolic if n is odd and Witt-equivalent to its restriction to ℓ_{m}λ^{n}M_{2}(B)ℓ_{m} if n = 2m.

We will also use the following:

Lemma 2.8. For any b ∈ F^{×}, consider g_{0} := ( \begin{matrix}0 & b \\ 1 & 0 \end{matrix} ) ∈ M_{2}(B), and set g := λ^{n}(g_{0}). We have:

1. for any u ∈ λ^{k}B, Φ(g)(u) = b^{n−k}γ_{k}(u) ∈ λ^{n−k}B;

2. g^{2} = b^{n} and γ(g) = (−1)^{n}g;

3. For any k = 0, …, n, gℓ_{k} = ℓ_{n−k}g.

Proof. Again, it is enough to prove it in the split case. A direct computation then shows that for any x ⊗ y ∈ ∧^{k}V ⊗ ∧^{n−k}V = λ^{k}B, we have

g(x ⊗ y) = (−1)^{k(n−k)}b^{n−k}(y ⊗ x),

which combined with (2.4) gives (1), which in turn easily implies (3). The first part of (2) is because λ^{n} restricts to be a group homomorphism on M_{2}(B)^{∗} [7, 14.3], and the second part then follows since γ(g)g = Nrd_{λ^{n}M_{2}(B)}(g) = (−b)^{n} by [7, 14.4].

3 Description of λ^{n}(Q ⊗ B)

We suppose that Q = (a, b)_{F} is a quaternion F -algebra and B is an arbitrary central simple F -algebra of degree n. We will describe λ^{n}(Q ⊗ B) by Galois descent from K = F (α), where α ∈ F_{s} is a fixed square root of a. More precisely, let us identify Q with the F -subalgebra of M_{2}(K) generated by ( \begin{matrix}α & 0 \\ 0 & −α \end{matrix} )

and g_{0} = ( \begin{matrix}0 & b \\ 1 & 0 \end{matrix} ), i.e.,

Q = {x ∈ M_{2}(K) | g_{0}x̄g_{0}^{−1} = x},

where ¯ denotes the non-trivial automorphism of K/F. We also have

Q ⊗ B = {x ∈ M_{2}(B_{K}) | g_{0}x̄g_{0}^{−1} = x},

where B_{K} = B ⊗_{F} K, and g_{0} is now viewed as an element of M_{2}(B_{K}). The canonical map λ^{n}: A → λ^{n}A restricts to be a group homomorphism on A^{∗} [7, 14.3]. Moreover, when deg A = 2n, for a ∈ A^{∗}, Int(λ^{n}(a)) preserves the canonical involution γ on λ^{n}A [7, 14.4], and so we get a map

λ^{n}: Aut(A) → Aut(λ^{n}A, γ).

In particular this holds for A = M_{2}(B_{K}). This induces a map on Galois cohomology

H^{1}(K/F, Aut(M_{2}(B_{K}))) ⟶^{H^{1}(λ^{n})} H^{1}(K/F, Aut(λ^{n}M_{2}(B_{K}), γ)).

The image under this map of the 1-cocycle ¯ ↦ Int(g_{0}) is the 1-cocycle ¯ ↦ Int(λ^{n}g_{0}), as in the preceding section. Since the former 1-cocycle corresponds to Q ⊗ B, the latter corresponds to λ^{n}(Q ⊗ B), so

λ^{n}(Q ⊗ B) = {x ∈ λ^{n}M_{2}(B_{K}) | gx̄g^{−1} = x} (3.1)

for g := λ^{n}(g_{0}). We fix this definition of g for the rest of the paper.

4 The n odd case

This section is essentially the proof of Theorem 1.1. We set λ^{even}B := ⊕\begin{matrix}0≤k<n \\ k even \end{matrix}λ^{k}B. For 0 ≤ k ≤ n, we let t_{k} be the reduced trace

quadratic form on λ^{k}B as in (1.2). We then have the following:

Lemma 4.1. When n = deg B is odd, the algebra with involution (λ^{n}(Q⊗B), γ) is isomorphic to (Q, γ_{Q}) ⊗ (C, σ), where (C, σ) is isomorphic to End_{F} (λ^{even}B) endowed with the adjoint involution with respect to ∑\begin{matrix}0≤k<n \\ k even \end{matrix} t_{k}.

Proof. If i, j ∈ Q satisfy i^{2} = a, j^{2} = b and ij = −ji, then since λ^{n} restricts to be a group homomorphism on (Q ⊗ B)^{∗}, λ^{n}(i ⊗ 1) and λ^{n}(j ⊗ 1) ∈ λ^{n}(Q ⊗ B) anticommute and satisfy

\begin{matrix}λ^{n}(i ⊗ 1)^{2} = a^{n}, & λ^{n}(j ⊗ 1)^{2} = b^{n}, \\ γ(λ^{n}(i ⊗ 1)) = −λ^{n}(i ⊗ 1), & γ(λ^{n}(j ⊗ 1)) = −λ^{n}(j ⊗ 1). \end{matrix}

(For the bottom two equations, see [7, (14.4)].) Hence, these two elements generate a copy of Q in λ^{n}(Q ⊗ B) on which γ restricts to be γ_{Q} and we have (λ^{n}(Q⊗B), γ) ≃ (Q, γ_{Q})⊗(C, σ), where C is the centralizer of Q in λ^{n}(Q⊗B) and σ denotes the restriction of γ to C [7, 1.5]. To describe C, we take i = α(e_{1} − e_{2}) and j = g_{0}, as in the beginning of the previous section, so that λ^{n}(j ⊗ 1) = g and

λ^{n}(i ⊗ 1) = α^{n}((−1)^{n}ℓ_{0} + (−1)^{n−1}ℓ_{1} + ⋯ + ℓ_{n}) = −α^{n}(ℓ_{even} − ℓ_{odd}),

where ℓ_{even} = ∑\begin{matrix}0≤k≤n \\ k even \end{matrix} ℓ_{k} and ℓ_{odd} = ∑\begin{matrix}0≤k≤n \\ k odd \end{matrix} ℓ_{k}.

Let us consider the map Ψ: ℓ_{even}λ^{n}(M_{2}(B))ℓ_{even} → λ^{n}(M_{2}(B_{K})) defined by Ψ(x) = x + gxg^{−1}. A direct computation shows that Ψ is an F -algebra homomorphism, amazingly. Clearly, \overline{Ψ(x)} = Ψ(x) and since g^{2} = b^{n} is central (see Lemma 2.8), gΨ(x) = Ψ(x)g for all x. Hence, the image of Ψ is contained in λ^{n}(Q ⊗ B) and is centralized by g. Moreover,

λ^{n}(i ⊗ 1)Ψ(x) = −α^{n}(x − gxg^{−1}) = Ψ(x)λ^{n}(i ⊗ 1).

Hence, the image of Ψ also centralizes λ^{n}(i ⊗ 1), and is therefore contained in C. Now, since ℓ_{even} is an idempotent of λ^{n}(M_{2}(B)), the algebra ℓ_{even}λ^{n}(M_{2}(B))ℓ_{even} is simple, hence Ψ is injective. By dimension count it follows that its image is exactly C. Since γ(Ψ(x)) = Ψ(g^{−1}γ(x)g), the involution σ on C corresponds via Ψ to Int(g^{−1}) ∘ γ on ℓ_{even}λ^{n}(M_{2}(B))ℓ_{even}. Note that if x ∈ ℓ_{even}λ^{n}(M_{2}(B))ℓ_{even}, then γ(x) ∈ ℓ_{odd}λ^{n}(M_{2}(B))ℓ_{odd} and g^{−1}γ(x)g ∈ ℓ_{even}λ^{n}(M_{2}(B))ℓ_{even}. By Theorem 2.5, we get that (C, σ) is isomorphic to End_{F} (λ^{even}B) endowed with the involution adjoint to the quadratic form T′ defined by T′(u, v) = T (u, Φ(g)(v)). Using the description of T given in Theorem 2.5 and Lemma 2.8(1), it is easy to check that the λ^{k}B are pairwise orthogonal for T′ and that T′ restricts to be ⟨(−b)^{n−k}⟩t_{k} on λ^{k}B. Thus T′ is similar to

∑\begin{matrix}0≤k<n \\ k even \end{matrix} t_{k}.

Let us now prove Theorem 1.1. If n = 2m+1, then the algebra with involution (Q, γ_{Q})^{⊗n} is isomorphic to (Q, γ_{Q}) ⊗ (End_{F} (Q), ad_{n_{Q}})^{⊗m}, where ad_{n_{Q}} denotes the adjoint involution with respect to the quadratic form n_{Q}. Indeed, one may easily check that (Q⊗Q, γ_{Q}⊗γ_{Q}) is isomorphic to (End_{F} (Q), ad_{T_{(Q,γ_{Q})}}), where T_{(Q,γ_{Q})} is the quadratic form defined by T_{(Q,γ_{Q})}(x) = Trd_{Q}(xγ_{Q}(x)). Since for any x ∈ Q, we have xγ_{Q}(x) = n_{Q}(x) ∈ F, T_{(Q,γ_{Q})} = ⟨2⟩n_{Q}, and (Q^{⊗2}, γ_{Q}^{⊗2}) ≃ (End_{F} (Q), ad_{n_{Q}}). Therefore, to prove Theorem 1.1, it suffices to show that the algebras with involution (Q, γ_{Q})⊗(C, σ) and (Q, γ_{Q})⊗(End_{F} (Q), ad_{n_{Q}})^{⊗m} are Witt-equivalent. We will use the following lemma:

Lemma 4.2. Let (U, q) and (U′, q′) be two quadratic spaces over F. There exists an isomorphism

(Q, γ_{Q}) ⊗ (End_{F} (U), ad_{q}) ≃ (Q, γ_{Q}) ⊗ (End_{F} (U′), ad_{q′})

if and only if the quadratic forms n_{Q} ⊗ q and n_{Q} ⊗ q′ are similar.

Proof. Consider the right Q-vector space U_{Q} = U ⊗_{F} Q. The quadratic form q on U induces a hermitian form h: U_{Q} × U_{Q} → Q (with respect to γ_{Q}) such that

h(u ⊗ x, u′ ⊗ x′) = \frac{1}{2}(q(u + u′) − q(u) − q(u′))γ_{Q}(x)x′

for u, u′ ∈ U and x, x′ ∈ Q. The adjoint involution ad_{h} satisfies

(4.3) (End_{Q}(U_{Q}), ad_{h}) = (End_{F} (U), ad_{q}) ⊗ (Q, γ_{Q}).

The trace form of h, which is by definition the quadratic form

U ⊗_{F} Q → F, x ↦ h(x, x),

is q ⊗ n_{Q}. Similarly, we denote by h′ the hermitian form induced by q′. By a theorem of Jacobson [13, 10.1.7], the hermitian modules (U_{Q}, h) and (U′_{Q}, h′) are isomorphic if and only if their trace forms are isometric. Hence, if the quadratic forms q ⊗ n_{Q} and q′ ⊗ n_{Q} are similar, i.e., q ⊗ n_{Q} ≃ ⟨μ⟩q′ ⊗ n_{Q} for some μ ∈ F^{∗}, then the hermitian forms h and ⟨μ⟩h′ are isomorphic, which proves that

(Q, γ_{Q}) ⊗ (End_{F} (U), ad_{q}) ≃ (Q, γ_{Q}) ⊗ (End_{F} (U′), ad_{q′}).

Conversely, if there is such an isomorphism, then equation (4.3) shows that the hermitian forms h and h′ are similar, hence their trace forms q ⊗ n_{Q} and q′ ⊗ n_{Q} also are similar.

These two lemmas reduce the proof of Theorem 1.1 to showing that the quadratic forms n_{Q} ⊗ ∑\begin{matrix}0≤k<n \\ k even \end{matrix} t_{k} and n_{Q}^{⊗(m+1)} are Witt-equivalent, up to a

scalar factor. On the one hand, we have n_{Q}^{⊗(m+1)} = 4^{m}n_{Q}, since n_{Q}^{⊗2} = 4n_{Q}. On the other hand, since the algebra B is split by an odd-degree field extension, Springer’s Theorem [8, VII.2.3] shows that t_{k} is isometric to the trace form of

λ^{k}(M_{n}(F)) = M\binom{n}{k}(F)

which is Witt-equivalent to \binom{n}{k}⟨1⟩. Hence the Witt class of n_{Q} ⊗ ∑\begin{matrix}0≤k<n \\ k even \end{matrix} t_{k} is

∑\begin{matrix}0≤k<n \\ k even \end{matrix} \binom{n}{k}n_{Q} = 2^{n−1}n_{Q} = 4^{m}n_{Q},

which completes the proof of Theorem 1.1.

5 The n even case

In this section, we prove Theorems 1.3, 1.4, and Corollary 1.5.

Assume from now on that n is even and write n = 2m. Consider the element of λ^{n}(M_{2}(B_{K}))

h = α(1 − b^{−m}g)(ℓ_{0} + ⋯ + ℓ_{m−1} + \frac{1}{2}ℓ_{m}) + (1 + b^{−m}g)(\frac{1}{2}ℓ_{m} + ℓ_{m+1} + ⋯ + ℓ_{n}).

One can check that

h^{−1} = \frac{1}{2} ((α^{−1} + b^{−m}g)(ℓ_{0} + ⋯ + ℓ_{m}) + (1 − b^{−m}gα^{−1})(ℓ_{m} + ⋯ + ℓ_{n}))

and g = b^{m} h\overline{h}^{−1}. Therefore, it follows from (3.1) that

λ^{n}(Q ⊗ B) = hλ^{n}M_{2}(B)h^{−1} ⊂ λ^{n}M_{2}(B)_{K}.

Using the isomorphism Φ of Theorem 2.5 as an identification, we then have

λ^{n}(Q ⊗ B) = End_{F} (h(λ^{0}B) ⊕ ⋯ ⊕ h(λ^{n}B)) ,

and the canonical involution on λ^{n}(Q ⊗ B) is adjoint to the restriction of the bilinear form T_{K} to the F -subspace h(λ^{0}B) ⊕ ⋯ ⊕ h(λ^{n}B). This restriction is given by the following formula:

Lemma 5.1. The F -subspaces h(λ^{k}B) are pairwise orthogonal. Moreover, for u, v ∈ λ^{k}B we have

T_{K} (h(u), h(v)) =

−2a(−1)^{k}b^{m−k} Trd_{λ^{k}B}(uv)  if k < m,
(−1)^{m}(\frac{1+a}{2} Trd_{λ^{m}B}(γ_{m}(u)v) + \frac{1−a}{2} Trd_{λ^{m}B}(uv)) if k = m,
2(−1)^{k}b^{m−k} Trd_{λ^{k}B}(uv)  if k > m.

Proof. Using Lemmas 2.6 and 2.8(1), one may easily check that for any u ∈ λ^{k}B, we have

h(u) = \begin{cases}α(u − b^{m−k}γ_{k}(u)) & if k < m, \\ \frac{1}{2}[(1 + α)u + (1 − α)γ_{k}(u)] & if k = m, \\ u + b^{m−k}γ_{k}(u) & if k > m. \end{cases}

The claim then follows from the description of T given in Theorem 2.5 and Lemma 2.8(1) by some direct computations. For instance, if u, v ∈ λ^{m}B, we get

T_{K}(h(u), h(v)) = (−1)^{m} Trd_{λ^{m}B_{K}} [h(u)γ_{m}(h(v))] (5.2)

by Theorem 2.5, and

\begin{matrix}h(u)γ_{m}(h(v)) = \\ \frac{1}{4}[(1 + α)^{2}uγ_{m}(v) + (1 − a)(uv + γ_{m}(u)γ_{m}(v)) + (1 − α)^{2}γ_{m}(u)v]. \end{matrix} (5.3)

Since Trd_{λ^{m}B}(uγ_{m}(v)) = Trd_{λ^{m}B}(γ_{m}(u)v) and Trd_{λ^{m}B}(γ_{m}(u)γ_{m}(v)) = Trd_{λ^{m}B}(uv), it follows that

Trd_{λ^{m}B_{K}} [(1 + α)^{2}uγ_{m}(v) + (1 − α)^{2}γ_{m}(u)v] = 2(1 + a) Trd_{λ^{m}B}(γ_{m}(u)v)

and

Trd_{λ^{m}B_{K}} [(1 − a)(uv + γ_{m}(u)γ_{m}(v))] = 2(1 − a) Trd_{λ^{m}B}(uv).

Therefore, (5.2) and (5.3) yield

T_{K}(h(u), h(v)) = (−1)^{m} \frac{1 + a}{2} Trd_{λ^{m}B}(γ_{m}(u)v) + (−1)^{m} \frac{1 − a}{2} Trd_{λ^{m}B}(uv).

This lemma provides a first description of the similarity class of q_{A}:

Proposition 5.4. If n is even, the similarity class of q_{A} contains the quadratic form:

(⊕_{0≤k<m}⟨2(−1)^{k}b^{m−k}⟩⟨1, −a⟩t_{k}) ⊕ ⟨(−1)^{m}⟩(t_{m}^{+} ⊕ ⟨−a⟩t_{m}^{−}).

Proof. Since the anti-isomorphism γ_{k} defines an isometry t_{k} ≃ t_{n−k}, the restriction of T_{K} to h(λ^{k}B ⊕ λ^{n−k}B), for all k < m, is

⟨2(−1)^{k}b^{m−k}⟩⟨1, −a⟩t_{k}.

Moreover, we have

\frac{1 + a}{2} Trd_{λ^{m}B}(γ_{m}(u)v) + \frac{1 − a}{2} Trd_{λ^{m}B}(uv) =

\begin{cases}Trd_{λ^{m}B}(uv) & if u ∈ Sym(λ^{m}B, γ_{m}), \\ −a Trd_{λ^{m}B}(uv) & if u ∈ Skew(λ^{m}B, γ_{m}). \end{cases}

Hence, the proposition clearly follows from the lemma.

5.5. Proof of Theorem 1.3. Theorem 1.3 is a consequence of the preceding results in the special case where Q = (a, b)_{F} is split. In that case, we may take b = 1 so that the matrix g_{0} = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) then decomposes as g_{0} = f_{0}f̄_{0}^{−1}, where f_{0} = ( \begin{matrix}1 & −α \\ 1 & α \end{matrix} ). Hence, if we let f = λ^{n}f_{0}, we have g = ff̄^{−1}. On the other hand, we also have g = hh̄^{−1}, for h as in the preceding section, hence f^{−1}h = \overline{f^{−1}h}, which means that f^{−1}h ∈ λ^{n}(M_{2}(B)). Considering the isomorphism Φ of Theorem 2.5 as an identification as we did in the preceding section, we get that f^{−1}h ∈ End_{F} (λ^{0}B ⊕ ⋯ ⊕ λ^{n}B), hence

h(λ^{0}B ⊕ ⋯ ⊕ λ^{n}B) = f(λ^{0}B ⊕ ⋯ ⊕ λ^{n}B).

To prove Theorem 1.3, we compute the restriction of T_{K} to this F -subspace in two different ways. First, we use [7, (14.4)], which says that f is a similarity for T_{K} with similarity factor Nrd_{M_{2}(B_{K})}(f_{0}) = (−2α)^{n} = 2^{n}a^{m}. Hence, for any u, v ∈ λ^{0}B ⊕ ⋯ ⊕ λ^{n}B, we have

T_{K}(f(u), f(v)) = 2^{n}a^{m}T (u, v).

By Remark 2.7 and Theorem 2.5, the form T is Witt-equivalent to its restriction to λ^{m}B, which is isometric to ⟨(−1)^{m}⟩(t_{m}^{+} ⊕ ⟨−1⟩t_{m}^{−}). Second, the restriction of T_{K} to h(λ^{0}B ⊕ ⋯ ⊕ λ^{n}B) has been computed in Lemma 5.1 and the proof of Proposition 5.4. Comparing the results, we get that the quadratic forms

(⊕_{0≤k<m}⟨2(−1)^{k}⟩⟨1, −a⟩t_{k}) ⊕ ⟨(−1)^{m}⟩(t_{m}^{+} ⊕ ⟨−a⟩t_{m}^{−})

and

⟨2^{n}a^{m}⟩⟨(−1)^{m}⟩(t_{m}^{+} ⊕ ⟨−1⟩t_{m}^{−})

are Witt-equivalent. If m is even, we get that the following equality holds in the Witt ring:

(∑_{0≤k<m}⟨2(−1)^{k}⟩⟨1, −a⟩t_{k}) + t_{m}^{+} + ⟨−a⟩t_{m}^{−} = t_{m}^{+} − t_{m}^{−},

from which we deduce

⟨1, −a⟩ ((∑_{0≤k<m} ⟨2(−1)^{k}⟩t_{k}) + t_{m}^{−}) = 0.

To finish the proof, we may assume a is an indeterminate over the base field F. The previous equality then implies that the quadratic form

(⨁_{0≤k<m} ⟨2(−1)^{k}⟩t_{k}) ⊕ t_{m}^{−}

is hyperbolic, which proves the theorem in this case. A similar argument finishes the proof for the m odd case.

Remark 5.6. Let t_{(λ^{m}B,γ_{m})} : λ^{m}B → F be the quadratic form

t_{(λ^{m}B,γ_{m})}(x) = Trd_{λ^{m}B}(γ_{m}(x)x).

Using Theorem 1.3, together with the facts that t_{n−k} = t_{k}, t_{(λ^{m}B,γ_{m})} = t_{m}^{+}−t_{m}^{−}, and that 2q ≃ 2⟨2⟩q for an arbitrary quadratic form q since 2⟨2⟩ = 2⟨1⟩, we obtain the following memorable formula:

∑_{k=0}^{n}(−1)^{k}t_{k} = t_{(λ^{m}B,γ_{m})} in W F .

5.7. Proof of Theorem 1.4. Consider first the case where m is even. In that case, Theorem 1.3 yields

∑\begin{matrix}0≤k<m \\ k even \end{matrix}⟨2⟩t_{k} + t_{m}^{−} = ∑\begin{matrix}0≤k<m \\ k odd \end{matrix}⟨2⟩t_{k}.

Substituting in the formula given in Proposition 5.4, we get that the similarity class of q_{A} contains a quadratic form whose Witt class is

∑\begin{matrix}0≤k<m \\ k even \end{matrix} ⟨2, −2a⟩t_{k} + ∑\begin{matrix}0≤k<m \\ k even \end{matrix} ⟨−2b, 2ab⟩t_{k} + ⟨−a, −b, ab⟩t_{m}^{−} + t_{m}^{+}

= ∑\begin{matrix}0≤k<m \\ k even \end{matrix} ⟨2⟩n_{Q}t_{k} + t_{m}^{+} − t_{m}^{−} + n_{Q}t_{m}^{−}.

Now, suppose m is odd. Multiplying by ⟨a⟩ the quadratic form given in Proposition 5.4 does not change its similarity class, and shows that the similarity class of q_{A} contains a quadratic form whose Witt class is

⟨1, −a⟩ · (t_{m}^{+} + ∑_{0≤k<m}⟨2(−b)^{k+1}⟩t_{k}) + t_{m}^{−} − t_{m}^{+}.

Substituting for t_{m}^{+} the formula of Theorem 1.3 simplifies the expression in brackets to ⟨1, −b⟩ · (∑\begin{matrix}0≤k<m \\ k even \end{matrix}⟨2⟩t_{k}) and completes the proof.

5.8. Proof of Corollary 1.5. Let us assume that B is of exponent at most 2. Then, for any even k, the algebra λ^{k}B is split. Hence, its trace form t_{k} is Witt-equivalent to \binom{n}{k}. Since m is even, λ^{m}B is also split, and its canonical involution γ_{m} is adjoint to a quadratic form q_{B}. This form is only defined up to a scalar factor, but its square is defined up to isometry. Now [7, 11.4] gives relationships between q_{B} and the forms t_{m}^{+} and t_{m}^{−}:

t_{m}^{+} − t_{m}^{−} ≃ q_{B}^{2}  and  − t_{m}^{−} ≃ ⟨1/2⟩ ∧^{2} q_{B}.

Hence, by Theorem 1.4, the similarity class of q_{A} contains a form whose Witt class is

q_{B}^{2} + n_{Q}(⟨−2⟩(∧^{2}q_{B}) + ∑\begin{matrix}0≤k<m \\ k even \end{matrix} \binom{n}{k}⟨2⟩).

One may easily check that, since ⟨2, 2⟩ ≃ ⟨1, 1⟩ and q_{B} is even-dimensional, q_{B}^{2} ≃ ⟨2⟩q_{B}^{2}. Since we are concerned only with the similarity class of q_{A}, we may therefore forget the factors ⟨2⟩ throughout. Moreover, since m is even, ∑\begin{matrix}0≤k<m \\ k even \end{matrix} \binom{n}{k} = 2^{n−2} − \frac{1}{2} \binom{n}{m}, and Corollary 1.5 follows.

6 Another approach to the n even case

Let us decompose B = B_{0} ⊗ B_{1}, where deg B_{0} = 2m_{0} is a power of 2 and deg B_{1} = m_{1} is odd. We have m = m_{0}m_{1}, and m is even if and only if m_{0} > 1.

We write T_{0} for the trace form of B_{0}. Under the assumption that B_{0}^{⊗2} is split (which is automatic if m is odd), we will give a different characterization of q_{A} for A = Q ⊗ B than the one in Theorem 1.4. Corollaries 1.6 and 1.7 will follow from this.

Proposition 6.1. Suppose that B_{0}^{⊗2} is split. Then the similarity class of q_{A} contains a form whose Witt class is

2^{n−1} + \frac{2^{n−3}}{m_{0}} T_{0}(n_{Q} − 2) if m is even

and

2^{n−2}(n_{Q} − n_{B_{0}}) if m is odd.

(Note that B_{0} is a quaternion algebra if m is odd.)

This result is already known for m odd: If A is a biquaternion algebra (i.e., m = 1) it is [3, 6.2], and in general it follows from [3, 6.4] by a straightforward computation, using the fact that for any integer k ≥ 1, one has n_{Q}^{k} = 2^{2(k−1)}n_{Q}. However, the results from [3] make use of Clifford algebras, which seems a long way to go. So we include a direct proof. We start with a lemma.

Lemma 6.2. Suppose that B_{0}^{⊗2} is split. Then the quadratic form t_{k} is Wittequivalent to \binom{n}{k} if k is even and \frac{1}{2m_{0}} \binom{n}{k}T_{0} if k is odd. Moreover, we have:

t_{m}^{−} = \frac{2^{n−3}}{m_{0}} ⟨2⟩T_{0} − (2^{n−2} − \frac{1}{2} \binom{n}{m})⟨2⟩ if m is even,

and

t_{m}^{+} = 2^{n−2}⟨2⟩ − (2^{n−3} − \frac{1}{4} \binom{n}{m})⟨2⟩T_{0} if m is odd.

This lemma actually specifies t_{m}^{+} and t_{m}^{−} whatever the parity of m since in both cases t_{m} = t_{m}^{+} + t_{m}^{−}, and t_{m} is known.

Proof. Since B_{1} is split by an odd-degree field extension, Springer’s Theorem shows that t_{k} is isometric to the trace form of λ^{k}(B_{0} ⊗ M_{m_{1}}(F)). If k is even, this algebra is split, and the result is clear. If k is odd, the algebra is Brauerequivalent to B_{0}, hence isomorphic to M_{p}(F) ⊗ B_{0}, where p = \frac{1}{2m_{0}} \binom{n}{k}. The form of t_{k} for k odd then follows from the fact that the trace form of a tensor product of central simple algebras is isometric to the product of the trace forms of each factor. We have m = m_{0}m_{1}, and m is odd if and only if m_{0} = 1. Recall that

∑\begin{matrix}0≤k<m \\ k even \end{matrix} \binom{n}{k} = \begin{cases}2^{n−2} & if m is odd, \\ 2^{n−2} − \frac{1}{2} \binom{n}{m} & if m is even, \end{cases}

and

∑\begin{matrix}0≤k<m \\ k odd \end{matrix} \binom{n}{k} = \begin{cases}2^{n−2} − \frac{1}{2} \binom{n}{m} & if m is odd, \\ 2^{n−2} & if m is even. \end{cases}

The second part of the lemma then follows from Theorem 1.3 by a direct computation.

Let us now prove Proposition 6.1. Assume first that m is even. The preceding lemma yields

t_{m}^{−} + ∑\begin{matrix}0≤k<m \\ k even \end{matrix}⟨2⟩t_{k} = \frac{2^{n−3}}{m_{0}} ⟨2⟩T_{0}

and

t_{m}^{+} − t_{m}^{−} = \binom{n}{m} − 2t_{m}^{−} = 2^{n−1}⟨2⟩ − \frac{2^{n−2}}{m_{0}} ⟨2⟩T_{0} + \binom{n}{m}⟨1, −2⟩.

Since \binom{n}{m} is even, the last term on the right side vanishes, hence the quadratic form given by Theorem 1.4 is

⟨2⟩(2^{n−1} − \frac{2^{n−2}}{m_{0}} T_{0} + \frac{2^{n−3}}{m_{0}} n_{Q}T_{0}).

This finishes the m even case. Assume now that m is odd. Then, B_{0} is a quaternion algebra, and T_{0} = ⟨2⟩(2 − n_{B_{0}}). The preceding lemma yields

∑\begin{matrix}0≤k<m \\ k even \end{matrix}⟨2⟩t_{k} = 2^{n−2}⟨2⟩

and

t_{m}^{−} − t_{m}^{+} = \frac{1}{2} \binom{n}{m}T_{0} − 2t_{m}^{+} = \frac{1}{2} \binom{n}{m}T_{0} − 2^{n−1}⟨2⟩ + (2^{n−2} − \frac{1}{2} \binom{n}{m})⟨2⟩T_{0}.

If m = 1, then this reduces to t_{m}^{−} − t_{m}^{+} = −⟨2⟩n_{B_{0}}, and Theorem 1.4 gives the desired result. Otherwise, since m is odd and m ≥ 3, the integer 2^{n−2} − \frac{1}{2} \binom{n}{m} is even, by [7, (10.29)], hence (2^{n−2} − \frac{1}{2} \binom{n}{m})⟨2⟩ = 2^{n−2} − \frac{1}{2} \binom{n}{m} and the right side of the last displayed equation simplifies to yield

t_{m}^{−} − t_{m}^{+} = −2^{n−2}⟨2⟩n_{B_{0}}.

Therefore, the quadratic form given by Theorem 1.4 is 2^{n−2}⟨2⟩(n_{Q} − n_{B_{0}}), which is isometric to 2^{n−2}(n_{Q} − n_{B_{0}}) since 2^{n−2}⟨2⟩ = 2^{n−2}, and the proof of Proposition 6.1 is complete. 6.3. Proof of Corollary 1.6. Corollary 1.6 can be proved by induction, using the formula given in Corollary 1.5, but it can also be directly deduced from Proposition 6.1. Indeed, let us assume A = A_{r} = Q_{1} ⊗ ⋯ ⊗ Q_{r} is a product of r ≥ 3 quaternion algebras. We let B = Q_{2} ⊗ ⋯ ⊗ Q_{r}. Its degree n = 2^{r−1} is a power of 2, and since r ≥ 3, m = 2^{r−2} is even. In the notation from earlier in this previous section, we have B_{0} = B and B_{0}^{⊗2} is split. Hence, we may apply Proposition 6.1. The form T_{0} is the trace form of B, that is the tensor product of the trace forms of the quaternion algebras Q_{i} for i = 2, …, r. Hence, we have T_{0} = ⟨2^{r−1}⟩(2 − n_{Q_{2}}) ⋯ (2 − n_{Q_{r}}), and Proposition 6.1 tells us that the similarity class of q_{A} contains a form whose Witt class is

\begin{matrix}2^{n−1} + \frac{2^{n−3}}{2^{r−2}} & ⟨2^{r−1}⟩(n_{Q_{1}} − 2)(2 − n_{Q_{2}}) ⋯ (2 − n_{Q_{r}}) = \\ = & 2^{n−1}⟨2^{r−1}⟩ − 2^{n−r−1}⟨2^{r−1}⟩(2 − n_{Q_{1}})(2 − n_{Q_{2}}) ⋯ (2 − n_{Q_{r}}) \\ = & ⟨2^{r−1}⟩2^{n−r−1}(2^{r} − (2 − n_{Q_{1}}) ⋯ (2 − n_{Q_{r}})), \end{matrix}

which proves the corollary.

6.4. Proof of Corollary 1.7. Let us now consider a central simple algebra A as in the statement of Corollary 1.7. Then A is isomorphic to M_{k}(A_{r}), where A_{r} = Q_{1} ⊗ ⋯ ⊗ Q_{r} is a product of r quaternion algebras. If A is split then q_{A} is hyperbolic and the result is clear, so we may assume that r ≠ 0. Because deg A ≡ 0 mod 4 by hypothesis, we may further assume that r ≠ 1 (so that r ≥ 2), with perhaps some of the Q_{i} being split. We first treat the k = 1 case. If r = 2, then A is biquaternion algebra and q_{A} is an Albert form, which lies in I^{2}F. If r ≥ 3, then by Corollary 1.6 we have to prove that

2^{n−1} − 2^{n−r−1}(2 − n_{Q_{1}}) ⋯ (2 − n_{Q_{r}})

lies in I^{n}F. When we expand this product, the terms of the form 2^{n−1} cancel, and we are left with a sum of terms of the form ±2^{n−ℓ−1}n_{Q_{i_{1}}} ⋯ n_{Q_{i_{ℓ}}}, where ℓ ≥ 1. Since for any i the form n_{Q_{i}} lies in I^{2}F, 2^{n−ℓ−1}n_{Q_{i_{1}}} ⋯ n_{Q_{i_{ℓ}}} belongs to I^{n−ℓ−1+2ℓ}F = I^{n+ℓ−1}F, and hence to I^{n}F. Now suppose that k ≥ 2. Since r ≥ 2, we have deg(A_{r}) ≡ 0 mod 4 and we can apply [3, 6.3(1)]. Hence, the similarity class of q_{A} contains a form which is Witt-equivalent to q_{A_{r}}^{⊗k}. Since the result holds for A_{r} by the k = 1 case, we are done.

Acknowledgments

The first author would like to thank ETH Zürich for its hospitality while some of the research for this paper was conducted. The second and third authors are partially supported by the European Commission under the TMR contract ERB-FMRX-CT97-0107. The third author is partially supported by the National Fund for Scientific Research (Belgium).

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R. Skip Garibaldi UCLA, Dept. of Mathematics, Los Angeles, CA 90095-1555, USA skip@member.ams.org http://www.math.ucla.edu/˜ skip/ Anne Quéguiner-Mathieu UMR CNRS 7539, Département de Mathématiques, Université Paris 13, F-93430 Villetaneuse, France queguin@math.univ-paris13.fr

Jean-Pierre Tignol

Départment de Mathématiques, Université Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium tignol@agel.ucl.ac.be http://www.math.ucl.ac.be/tignol/