Chapter 8 Simple algebras and involutions In this chapter, we examine further connections between quaternion algebras, simple algebras, and involutions. 8.1 The Brauer group and involutions ∼ An involution on an F-algebra B induces an isomorphism : B −→ Bop, for example such an isomorphism is furnished by the standard involution on a quaternion algebra B. More generally, if B1, B2 are quaternion algebras, then the tensor product B1 ⊗F B2 has an involution provided by the standard involution on each factor giving an isomorphism B ⊗ B op Bop ⊗ Bop to ( 1 F 2) 1 F 2 —but this involution is no longer a standard involution (Exercise 8.1). The algebra B1 ⊗F B2 is a central simple algebra over F called a biquaternion algebra. In some circumstances, we may have B1 ⊗F B2 M2(B3) (8.1.1) where B3 is again a quaternion algebra, and in other circumstances, we may not; following Albert, we begin this chapter by studying (8.1.1) and biquaternion algebras in detail. To this end, we look at the set of isomorphism classes of central simple algebras over F, which is closed under tensor product; if we think that the matrix ring is something that is ‘no more complicated than its base ring’, it is natural to introduce an equivalence relation on central simple algebras that identifies a division ring with the matrix ring (of any rank) over this division ring. More precisely, if A, A are central simple algebras over F we say A, A are Brauer equivalent if there exist n, n ≥ 1such that Mn(A) Mn (A ). In this way, (8.1.1) reads B1 ⊗F B2 ∼ B3. The set of Brauer equivalence classes [A] has the structure of a group under tensor product, known as the Brauer group Br(F)ofF, with identity element [F] and inverse [A]−1 = [Aop]. The class [B] ∈ Br(F) of a quaternion algebra B is a 2-torsion element, and therefore so is a biquaternion algebra. In fact, by a striking theorem of Merkurjev, when char F 2, all 2-torsion elements in Br(F) are represented by a tensor product of quaternion algebras (see section 8.3). © The Author(s) 2021 123 J. Voight, Quaternion Algebras, Graduate Texts in Mathematics 288, https://doi.org/10.1007/978-3-030-56694-4_8 124 CHAPTER 8. SIMPLE ALGEBRAS AND INVOLUTIONS Finally, our interest in involutions in Chapter 3 began with an observation of Hamilton: the product of a nonzero element with its involute in H is a positive real number (its norm, or square length). We then proved that the existence of such an involution characterizes quaternion algebras in an essential way. However, one may want to relax this setup and instead consider when the product of a nonzero element with its involute merely has positive trace. Such involutions are called positive involutions and they arise naturally in algebraic geometry: the Rosati involution is a positive involution on the endomorphism algebra of an abelian variety, and it is a consequence that this algebra (over Q) is semisimple, and unsurprisingly quaternion algebras once again feature prominently (see sections 8.4–8.5). 8.2 Biquaternion algebras Let F be a field. All tensor products in this section will be taken over F. 8.2.1. Let B1, B2 be quaternion algebras over F. The tensor product B1 ⊗ B2 is a central simple algebra over F of dimension 42 = 16 called a biquaternion algebra.A biquaternion algebra may be written as a tensor product of two quaternion algebras in different ways, so the pair is not intrinsic to the biquaternion algebra. By the Wedderburn–Artin theorem (Main Theorem 7.3.10), we have exactly one of the three following possibilities for this algebra: • B1 ⊗ B2 is a division algebra; • B1 ⊗ B2 M2(B3) where B3 is a quaternion division algebra over F;or • B1 ⊗ B2 M4(F). We could combine the latter two and just say that B1 ⊗ B2 M2(B3) where B3 is a quaternion algebra over F, since M2(M2(F)) M4(F)asF-algebras. Example 8.2.2. By Exercise 8.2, when char F 2wehave a, b a, b 1 ⊗ 2 B F F M2( 3) a, b b a, b a, b a, b2 B = 1 2 ⊗ F F where 3 F . In particular, F F M4( ), since F M2( ). Example 8.2.2 is no accident, as the following proposition indicates. Proposition 8.2.3. (Albert). The following are equivalent: (i) There exists a quadratic field extension K ⊃ F that can be embedded as an F-algebra in both B1 and B2; (ii) B1 and B2 have a common quadratic splitting field; and (iii) B1 ⊗ B2 is not a division algebra. Proof. The equivalence (i) ⇔ (ii) follows from Lemma 5.4.7. 8.2. BIQUATERNION ALGEBRAS 125 ⇒ i = , α ∈ B K α2 = tα − n For the implication (i) (iii), for 1 2let i i generate so i i with t, n ∈ F.Let β := α1 ⊗ 1 − 1 ⊗ α2. Then 2 2 β(α1 ⊗ 1 + 1 ⊗ α2 − t) = α ⊗ 1 − 1 ⊗ α − tβ 1 2 (8.2.4) = (tα1 − n) ⊗ 1 − 1 ⊗ (tα2 − n) − tβ = 0. Therefore β is a zerodivisor and B1 ⊗ B2 is not a division algebra. To finish, we prove (iii) ⇒ (i). We have an embedding B1 → B1 ⊗ B2 α → α ⊗ 1 and similarly B2; the images of B1 and B2 in B1 ⊗ B2 commute. Write B2 = (K, b2 | F). Consider (B1)K = B1 ⊗ K ⊂ B1 ⊗ B2; then (B1)K is a quaternion algebra over K (with dimF (B1)K = 8). If (B1)K is not a division algebra, then K splits B1 and K → B1 and we are done. So suppose that (B1)K is a division algebra. Then B1 ⊗ B2 = (B1)K + (B1)K j is free of rank 2 as a left (B1)K -module. Since B1 ⊗ B2 M2(B3) is not a division algebra, there exists ∈ B1 ⊗ B2 nonzero 2 such that = 0. Without loss of generality, we can write = α1 ⊗ z + j where α1 ∈ B1 and z ∈ K. Then = 2 = α2 ⊗ z2 + α ⊗ z j + α ⊗ z j + b . 0 1 ( 1 ) ( 1 ) 2 (8.2.5) From the basis 1, j over (B1)K ,ifz = t − z with t ∈ F, we conclude that α1 ⊗ z + α1 ⊗ (t − z) = α1 ⊗ t = 0. t = z2 = c c ∈ F× cα2 + b = Therefore 0, and for some . Then√ from (8.2.5) 1 2 0 α2 = −b /c B F −b c B so 1 2 and 1 contains the quadratic field ( 2 ). But so does 2,as 2 (zj) = −b2c as well. (For an alternate proof, see Jacobson [Jacn2009, Theorem 2.10.3].) Remark 8.2.6. In view of Proposition 8.2.3, we say that two quaternion algebras B1, B2 over F are linked if they contain a common quadratic field extension K ⊇ F. For further discussion of biquaternion algebras and linkage in characteristic 2 (where one must treat separable and inseparable extensions differently), see Knus [Knu93], Lam [Lam2002], or Sah [Sah72]. Garibaldi–Saltman [GS2010] study the subfields of quaternion algebra over fields with char F 2. From now on, we suppose that char F 2. (For the case char F = 2, see Chapman– Dolphin–Laghbribi [CDL2015, §6].) 126 CHAPTER 8. SIMPLE ALGEBRAS AND INVOLUTIONS 8.2.7. Motivated by Proposition 8.2.3, we consider the quadratic extensions repre- sented by B1 and B2 encoded in the language of quadratic forms (recalling Lemma 5.5.4). Let V = {α1 ⊗ 1 − 1 ⊗ α2 ∈ B1 ⊗ B2 :trd(α1) = trd(α2)}. V = V = B0 ⊗ − ⊗ B0 Then dimF 6, and we may identify 1 1 1 2. The reduced norm on each factor separately defines a quadratic form on V by taking the difference: explicitly, if B1 = (a1, b1 | F) and B2 = (a2, b2 | F), then taking the standard bases for B1, B2 Q(B1, B2) −a1, −b1, a1b1 −−a2, −b2, a2b2 −a1, −b1, a1b1, a2, b2, −a2b2. The quadratic form Q(B1, B2):V → F is called the Albert form of the biquaternion algebra B1 ⊗ B2. We then add onto Proposition 8.2.3 as follows. Proposition 8.2.8. (Albert).LetB1 ⊗ B2 be a biquaternion algebra over F (with char F 2) with Albert form Q(B1, B2). Then the following are equivalent: (i) B1, B2 have a common quadratic splitting field; (iv) Q(B1, B2) is isotropic. Proof. The implication (ii) ⇒ (iv) follows by construction 8.2.7. To prove (iv) ⇒ (ii), without loss of generality, we may suppose B1, B2 are division algebras; then Q α ∈ B α ∈ B an isotropic vector of corresponds√ to elements 1 1 and 2 2 such that α2 = α2 = c ∈ F× K = F c 1 2 . Therefore ( ) is a common quadratic splitting field. Remark 8.2.9. Albert’s book [Alb39] on algebras still reads well today. The proof of Proposition 8.2.3 is due to him [Alb72]. (“I discovered this theorem some time ago. There appears to be some continuing interest in it, and I am therefore publishing it now.”) Albert used Proposition 8.2.8 to show that − , − x, B = 1 1 B = y 1 F and 2 F over F = R(x, y) have tensor product B1 ⊗F B2 a division algebra by verifying that the Albert form Q(B1, B2) is anisotropic over F. See Lam [Lam2005, Example VI.1] for more details. For the fields of interest in this book (local fields and global fields), a biquaternion algebra will never be a division algebra—the proof of this fact rests on classification results for quaternion algebras over these fields, which we will take up in earnest in Part II.