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Simple Algebras Chapter 7 Simple algebras In this chapter, we return to the characterization of quaternion algebras. We initially defined quaternion algebras in terms of generators and relations in Chapter 2;inthe chapters that followed, we showed that quaternion algebras are equivalently noncom- mutative algebras with a nondegenerate standard involution. Here, we pursue another approach, and we characterize quaternion algebras in a different way, as central simple algebras of dimension 4. 7.1 Motivation and summary Consider now the “simplest” sorts of algebras. Like the primes among the integers or the finite simple groups among finite groups, it is natural to seek algebras that cannot be “broken down” any further. Accordingly, we say that a ring A is simple if it has no nontrivial two-sided (bilateral) ideals, i.e., the only two-sided ideals are {0} and A.To show the power of this notion, consider this: if φ: A → A is a ring homomorphism and A is simple, then φ is either injective or the zero map (since ker φ ⊆ B is a two-sided ideal). A division ring A is simple, since every nonzero element is a unit and therefore every nonzero ideal (left, right, or two-sided) contains 1 so is equal to A. In particular, a field is a simple ring, and a commutative ring is simple if and only if it is a field. The matrix ring Mn(F) over a field F is also simple, something that can be checked directly by multiplying by matrix units (Exercise 7.5). Moreover, quaternion algebras are simple. The shortest proof of this statement, given what we have done so far, is to employ Main Theorem 5.4.4 (and Theorem 6.4.11 in characteristic 2): a quaternion algebra B over F is either isomorphic to M2(F) or is a division ring, and in either case is simple. One can also prove this directly (Exercise 7.1). Although the primes are quite mysterious and the classification of finite simple groups is a monumental achievement in group theory, the situation for algebras is quite simple, indeed! Our first main result is as follows (Main Theorem 7.3.10). © The Author(s) 2021 95 J. Voight, Quaternion Algebras, Graduate Texts in Mathematics 288, https://doi.org/10.1007/978-3-030-56694-4_7 96 CHAPTER 7. SIMPLE ALGEBRAS Theorem 7.1.1. (Wedderburn–Artin).LetF be a field and B be a finite-dimensional F-algebra. Then B is simple if and only if B Mn(D) where n ≥ 1 and D is a finite-dimensional division F-algebra. As a corollary of Theorem 7.1.1, we give another characterization of quaternion algebras. Corollary 7.1.2. Let B be an F-algebra. Then the following are equivalent: (i) B is a quaternion algebra; al al al (ii) B ⊗F F M2(F ), where F is an algebraic closure of F; and (iii) B is a central simple algebra of dimension dimF B = 4. Moreover, a central simple algebra B of dimension dimF B = 4 is either a division algebra or has B M2(F). This corollary has the neat consequence that a division algebra B over F is a quaternion algebra over F if and only if it is central of dimension dimF B = 4. For the reader in a hurry, we now give a proof of this corollary without invoking the Wedderburn–Artin theorem; this proof also serves as a preview of some of the ideas that go into the theorem. Proof of Corollary 7.1.2. The statement (i) ⇒ (ii) was proven in Exercise 2.4(d). al al al To prove (ii) ⇒ (iii), suppose B is an algebra with B := B ⊗F F M2(F ). The Fal-algebra Bal is central simple, from above. Thus Z(B) = Z(Bal) ∩ B = F.AndifI al al al al is a two-sided ideal of B then I := I ⊗F F is a two-sided ideal of B ,soI = {0} Ial = Bal I = Ial ∩ F B = Bal = or is trivial, whence is trivial. Finally, dimF dimFal 4. Finally, we prove (iii) ⇒ (i). Let B a central simple F-algebra of dimension 4. If B is a division algebra we are done; so suppose not. Then B has a nontrivial left ideal (e.g., one generated by a nonunit); let {0} I B be a nontrivial left ideal with 0 < m = dimF I minimal. Then there is a nonzero homomorphism B → EndF (I) Mm(F) which is injective, since B is simple. By dimension, we cannot have m = 1; if m = 2, then B M2(F) and we are done. So suppose m = 3. Then by minimality, every nontrivial left ideal of B has dimension 3. But for any α ∈ B, we have that Iα is a left ideal, so the left ideal I ∩ Iα is either {0} or I; in either case, Iα ⊆ I and I is a right ideal as well. But this contradicts the fact that B is simple. The Wedderburn–Artin theorem is an important structural result used throughout mathematics, so we give in this chapter a self-contained account of its proof. More generally, it will be convenient to work with semisimple algebras, finite direct products of simple algebras. When treating ideals of an algebra we would be remiss if we did not discuss more generally modules over the algebra, and the notions of simple and semisimple module are natural concepts in linear algebra and representation theory: a semisimple module is one that is a direct sum of simple modules (“completely reducible”), analogous to a semisimple operator where every invariant subspace has an invariant complement (e.g., a diagonalizable matrix). The second important result in this chapter is a theorem that concerns the simple subalgebras of a simple algebra, as follows (Main Theorem 7.7.1). 7.2. SIMPLE MODULES 97 Theorem 7.1.3. (Skolem–Noether).LetA, B be simple F-algebras and suppose that B is central. Suppose that f, g : A → B are homomorphisms. Then there exists β ∈ B such that f (α) = β−1g(α)β for all α ∈ A. Corollary 7.1.4. Every F-algebra automorphism of a simple F-algebra B is inner, i.e., Aut(B) B×/F×. Just as above, for our quaternionic purposes, we can give a direct proof. Corollary 7.1.5. Let B be a quaternion algebra over F and let K , K ⊂ B be quadratic ∼ 1 2 subfields. Suppose that φ: K1 −→ K2 is an isomorphism of F-algebras. Then φ lifts to −1 an inner automorphism of B, i.e., there exists β ∈ B such that α2 = φ(α1) = β α1 β −1 for all α1 ∈ K1. In particular, K2 = β K1 β. Proof. Write K1 = F(α1) with α1 ∈ B and let α2 = φ(α1) ∈ K2 ⊂ B,soK2 = F(α2). × −1 We want to find β ∈ B such that α2 = β α1 β. In the special case B M2(F), then α1,α2 ∈ M2(F) satisfy the same irreducible characteristic polynomial, so by the theory −1 × of rational canonical forms, α2 = β α1 β where β ∈ B GL2(F) as desired. Suppose then that B is a division ring. Then the set W = {β ∈ B : βα2 = α1 β} (7.1.6) is an F-vector subspace of B.LetFsep be a separable closure of F. (Or, apply Exercise 6.5 and work over a splitting field K linearly disjoint from K1 K2.) Then we have sep sep B ⊗F F M2(F ), and the common characteristic polynomial of α1,α2 either remains irreducible over Fsep (if K ⊃ F is inseparable) or splits as the product of two linear factors with distinct roots. In either case, the theory of rational canonical forms sep sep again applies, and there exists β ∈ B ⊗F F GL2(F ) that will do; but then by sep × linear algebra dimFsep W ⊗F F = dimF W > 0, so there exists β ∈ B \{0} = B with the desired property. As shown in the above proof, Corollary 7.1.5 can be seen as a general reformulation of the rational canonical form from linear algebra. 7.2 Simple modules Basic references for this section include Drozd–Kirichenko [DK94, §1–4], Curtis– Reiner [CR81, §3], Lam [Lam2001, §2–3], and Farb–Dennis [FD93, Part I]. An ele- mentary approach to the Weddernburn–Artin theorem is given by Brešar [Bre2010]. An overview of the subject of associative algebras is given by Pierce [Pie82] and Jacobson [Jacn2009]. Throughout this chapter, let B be a finite-dimensional F-algebra. To understand the algebra B, we look at its representations. A representation of B (over F) is a vector space V over F together with an F-algebra homomorphism B → EndF (V). Equivalently, a representation is given by a left (or right) B-module V: this is almost a tautology. Although one can define infinite-dimensional representations, they will not interest us here, and we suppose throughout that dimF V < ∞,or 98 CHAPTER 7. SIMPLE ALGEBRAS equivalently that V is a finitely generated (left or right) B-module. If we choose a basis for V, we obtain an isomorphism EndF (V) Mn(F) where n = dimF V,soa representation is just a homomorphic way of thinking of the algebra B as an algebra of matrices. n Example 7.2.1. The space of column vectors F is a left Mn(F)-module; the space of row vectors is a right Mn(F)-module. Example 7.2.2. B is itself a left B-module, giving rise to the left regular representa- tion B → EndF (B) over F (cf.
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