Involutions and Trace Forms on Exterior Powers of a Central Simple Algebra

For A a central simple algebra of degree 2n, the n-th exterior power algebra λ^nA 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.

Associated to every central simple F -algebra A and any nonnegative integer k ≤ deg A is the kth exterior power λ k A of A, which is a central simple F -algebra, of degree deg A k , Brauer-equivalent to A ⊗k , see [4,10.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 ). When A has even degree 2n, 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 .
Besides providing an invariant of A, the involution γ is of additional interest because of the even Clifford algebra.Indeed, any central simple algebra A with hyperbolic orthogonal involution can be written as (M 2 (B), σ), for some central simple algebra B of degree n uniquely determined up to isomorphism.The Clifford algebra C M 2 (B), σ is itself an algebra with involution, and has been completely described when n is odd: in this case, M 2 (B) is the endomorphism algebra of an F -vector space V , and the Clifford algebra C M 2 (B), σ is the even Clifford algebra of any quadratic form on V with adjoint involution σ.However, in the case when n is even, C(M 2 (B), σ) has a nontrivial piece, which is isomorphic to (λ n M 2 (B), γ).Please see [2] for a precise statement and [7] for a rational proof.
If A has degree 2n for n even, then γ is of orthogonal type, and if moreover A ⊗n is split, then λ n A is split as well and hence γ is adjoint to some quadratic form q A .This provides a canonical way to associate to A a quadratic form q A of dimension 2n n , which is uniquely determined up to a scalar factor.If A is a biquaternion algebra, then q A is an Albert form for A [2, 6.2].Until now, the value of q A has not been known for any algebra A of index ≥ 8.
The main purpose of this paper is to provide a description of this involution γ in the particular case when A admits a decomposition A = Q ⊗ B, where Q is a quaternion algebra over F (see below for precise statements).In particular, if A is a tensor product of quaternion algebras, we get a formula that gives q A (up to Witt-equivalence) in terms of the norm forms of the quaternion algebras.In the course of obtaining this description, we also prove a formula relating the trace forms of the various exterior powers λ k B.
Date: May 15, 2000.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).

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.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: 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 Moreover, 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 The forms thus defined are related by the following equation, proven in 5.3: Theorem 1.3.In the Witt ring of F , the following equality holds: The similarity class of q A is determined by the following theorem, proven in 5.5: Theorem 1.4.If n is even, n = 2m, the similarity class of q A contains the quadratic form: 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 5.6: 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 . 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 Thus if q = α 1 , . . ., α n , we have In particular, even if q is just defined up to similarity, ∧ 2 q is well-defined up to isometry.
From Corollary 1.5, we also get the following, which is proven in 6.3: a tensor product of r quaternion F -algebras, where r ≥ 3, and let T Ar be the reduced trace quadratic form on A r .The similarity class of q Ar contains a quadratic form whose Witt class is where n = 2 r−1 = 1 2 deg A and f (r) = 2 r−1 − r − 1.In particular, for r = 3, we get the quadratic form Adrian Wadsworth had casually conjectured a description of q A3 in [2, 6.8], and we now see that his conjecture was not quite correct in that it omitted the n Q1 n Q2 n Q3 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 [2, 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 ′′ [3, 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 [4, (16.5)]).Corollary 1.6 shows that one cannot expect nice results relating the Witt index of q Ar 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 [6, §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 Ar are then hyperbolic.

Description of
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 2.4 below.
Assume B = End F (V ) for some n-dimensional vector space V .For 0 ≤ k ≤ n, we have 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 In particular, we have 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 This form is nonsingular, so it provides the identification mentioned above From (2.1) and (2.2), we deduce an identification of the corresponding endomorphism rings . 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 for x and y as before.One may easily verify that γ n−k •γ k = Id EndF (∧ 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 antiisomorphisms γ k : [4,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 2.4.There is a canonical isomorphism 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 On the other hand, the canonical action of Γ on B ⊗ F F s corresponds under ϕ to some twisted action * on End Fs (V s ).Since every F s -linear automorphism of End Fs (V s ) is inner, we may find g σ ∈ GL(V s ) such that Then ϕ induces an F -algebra isomorphism from B onto the F -subalgebra The * -action of Γ on End Fs (V s ) induces twisted actions on End Fs ∧ n (V s ⊕ V s ) and on End Fs ⊕ n k=0 End Fs (∧ k V s ) such that the F -algebras of Γ-invariant elements are λ n M 2 (B) and End F (⊕ n k=0 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

1), and computation shows that
2).Therefore, (2.1) and (2.2) induce an isomorphism of F -algebras 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 (⊕ n k=0 λ k B) as the fixed subalgebras of End Fs ∧ n (V s ⊕V s ) and End Fs ⊕ n k=0 End Fs (∧ k V s ) , and show that the canonical involution γ on End Fs ∧ 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 as was mentioned in the introduction.Using the identification of 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 Fs (∧ k V s ).In particular, we know that for Trd the reduced trace, and that Of course, in the k + ℓ = n case we could just as easily have taken Θ(x, y) = (−1) ℓ Trd λ ℓ Bs (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 = ( 1 0 0 0 ) and e 2 = ( 0 0 0 1 ) ∈ 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 [4, 14.3], which is a homogeneous polynomial map of degree n.So there exist ℓ 0 , . . ., ℓ n ∈ λ n M 2 (B) such that We then have Lemma 2.7.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 ( , 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: 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 2.4, to prove the second one, one has to check that for any u, v , which follows easily from the description of T given in that theorem.
Proof.Again, it is enough to prove it in the split case.A direct computation then shows that for any which combined with (2.3) 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) * [4, 14.3], and the second part then follows since γ(g)g = Nrd M2(B) (g) = (−b) n by [4, 14.4].

Description of λ
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 Fsubalgebra of M 2 (K) generated by α 0 0 −α and g 0 = ( 0 b 1 0 ), i.e., where ¯denotes the non-trivial automorphism of K/F .We also have , 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 * [4, 14.3].Moreover, when deg A = 2n, for a ∈ A * , Int(λ n (a)) preserves the canonical involution γ on λ n A [4, 14.4], and so we get a map In particular this holds for A = M 2 (B K ).This induces a map on Galois cohomology 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 for g := λ n (g 0 ).We fix this definition of g for the rest of the paper.

The n odd case
This section is essentially the proof of Theorem 1.1.We set λ even B := ⊕ 0≤k<n k even 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: where (C, σ) is isomorphic to End F (λ even B) endowed with the adjoint involution with respect to 0≤k<n k even 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 (For the bottom two equations, see [4, (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 ( where C is the centralizer of Q in λ n (Q ⊗ B) and σ denotes the restriction of γ to C [4, 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 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 .Believe it or not, Ψ is an F -algebra homomorphism.Clearly, Ψ(x) = Ψ(x) and since g 2 = b n is central (see Lemma 2.9), 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, Hence, the image of Ψ also centralizes λ n (i ⊗ 1), and by dimension count it 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 2.4, 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 2.4 and Lemma 2.9(1), it is easy to check that the λ k B are pairwise orthogonal for T ′ and that T ′ restricts to be ( , where ad nQ denotes the adjoint involution with respect to the quadratic form n Q .Indeed, one may easily check that . 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 nQ ) ⊗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 if and only if the quadratic forms n Q ⊗ q and n Q ⊗ q ′ are similar.
Proof.Let us write h for the hermitian form h : Its trace form, which is by definition the quadratic form is q ⊗ n Q .Similarly, we denote by h ′ the hermitian form induced by q ′ .By a theorem of Jacobson [8, 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 Conversely, if the two algebras with involution (Q, γ Q ) ⊗ (End F (U ), ad q ) and (Q, γ Q ) ⊗ (End F (U ′ ), ad q ′ ) are isomorphic, then, (Q, γ Q ) ⊗2 ⊗(End F (U ), ad q ) = (End F (Q⊗U ), ad nQ⊗q ) and (Q, γ Q ) ⊗2 ⊗ (End F (U ′ ), ad q ′ ) are also isomorphic, which proves that n Q ⊗ q and n Q ⊗ q ′ are similar.These two lemmas reduce the proof of Theorem 1.1 to showing that the quadratic forms n Q ⊗ 0≤k<n k even t k and n ⊗(m+1) Q are Witt-equivalent, up to a scalar factor.
On the one hand, we have n On the other hand, since the algebra B is split by an odd-degree field extension, Springer's Theorem [5,VII.2.3] shows that t k is isometric to the trace form of which completes the proof of Theorem 1.1.

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 One can check that Therefore, it follows from (3.1) that Using the isomorphism Φ of Theorem 2.4 as an identification, we then have and the canonical involution on λ n (Q ⊗ B) is adjoint to the restriction of the bilinear form 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 Proof.Using Lemmas 2.7 and 2.9(1), one may easily check that for any u ∈ λ k B, we have The claim then follows from the description of T given in Theorem 2.4 and Lemma 2.9(1) by some direct computations.For instance, if u, v ∈ λ m B, we get This lemma yields a first description of the similarity class of q A : Proposition 5.2.If n is even, the similarity class of q A contains the quadratic form: Moreover, we have Hence, the proposition clearly follows from the lemma.

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 = ( 0 1 1 0 ) then decomposes as g 0 = f 0 f −1 0 , where f 0 = 1 −α 1 α .Hence, if we let f = λ n f 0 , we have g = f f −1 .On the other hand, we also have g = h h−1 , for h as in the preceding section, hence f −1 h = f −1 h, which means that f −1 h ∈ λ n M 2 (B) .Considering the isomorphism Φ of Theorem 2.4 as an identification as we did in the preceding section, we get that To prove Theorem 1.3, we compute the restriction of T K to this F -subspace in two different ways.First, we use [4, (14.4)], which says that f is a similarity for T K with similarity factor Nrd M2(BK By Remark 2.8 and Theorem 2.4, 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 5.2.Comparing the results, we get that the quadratic forms If m is even, we get that the following equality holds in the Witt ring: 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 5.4.Let t (λ m B,γm) : λ m B → F be the quadratic form Using Theorem 1.3, together with the facts that t n−k = t k , t (λ m B,γm) = 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: in W F .
5.5.Proof of Theorem 1.4.Consider first the case where m is even.In that case, Theorem 1.3 yields Substituting in the formula given in Proposition 5.2, we get that the similarity class of q A contains a quadratic form whose Witt class is Now, suppose m is odd.Multiplying by a the quadratic form given in Proposition 5.2 does not change its similarity class, and shows that the similarity class of q A contains a quadratic form whose Witt class is Substituting for t + m the formula of Theorem 1.3 simplifies the expression in brackets to 1, −b • 0≤k<m k even 2 t k and completes the proof.5.6.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 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 [4, 11.4] gives relationships between q B and the forms t + m and t Hence, by Theorem 1.4, the similarity class of q A contains a form whose Witt class is One may easily check that, since 2, 2 ≃ 1, 1 and q B is even-dimensional, q 2 B ≃ 2 q 2 B .Since we are concerned only with the similarity class of q A , we may therefore forget the Let us now prove Proposition 6.1.Assume first that m is even.The preceding lemma yields Since n m is even, the last term on the right side vanishes, hence the quadratic form given by Theorem 1.4 is 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 B0 ).The preceding lemma yields 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 ⊗2 0 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 Q2 ) • • • (2 − n Qr ), and Proposition 6.1 tells us that the similarity class of q A contains a form whose Witt class is 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 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 Qi 1 • • • n Qi ℓ , where ℓ ≥ 1.Since for any i the right side simplifies to yield t − m − t + m = −2 n−2 n B0 .Therefore, the quadratic form given by Theorem 1.4 is 2 n−2 (n Q − n B0 ), and the proof of Proposition 6.1 is complete.