An introduction to quaternion algebras Benjamin Linowitz Department of Mathematics Oberlin College An Introduction to Quaternion Algebras One of the many nice features of the complex numbers is that their multiplication can be described using the geometric operations of stretching and rotation. The importance of the complex numbers in two-dimensional geometry was recognized very early. Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras William Rowan Hamilton sought a number system which would play an analogous role in three-dimensional geometry. \Every morning in the early part of the above-cited month [October 1843], on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me: Well, Papa, can you multiply triplets? Whereto I was always obliged to reply, with a sad shake of the head: No, I can only add and subtract them." { Hamilton (in a letter to his son) Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras Hamilton never succeeded in finding a three-dimensional real division algebra. In fact, we now know that no such algebra exists! Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras In more modern language, Hamilton was looking for an associative multiplication law on ∼ 3 D = R + Ri + Rj = R where i 2 = −1 and every nonzero element of D has an inverse. Suppose that there are a; b; c 2 R such that ij = a + bi + cj: If we multiply on the left by i we get −j = ai − b + cij. Substituting our formula for ij into this expression yields 0 = (ac − b) + (a + bc)i + (c2 + 1)j: 2 But this means that c = −1, a contradiction as c 2 R. Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras Hamilton did, however, find a four dimensional real division algebra. Theorem (Hamilton, 1843) The R-algebra H with basis f1; i; j; ijg and defining relations i 2 = −1 j2 = −1 ij = −ji is a four-dimensional division algebra. William Rowan Hamilton (1805-1865) Hamilton was so excited by this discovery that he carved these relations into the stone of the Brougham Bridge. Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras Modern vector analysis was pioneered by Willard Gibbs. In his treatise Elements of Vector Analysis he introduced the now standard notation of dot products and cross products, and related them to quaternionic multiplication via the beautiful formula vw = −v · w + v × w; where v; w 2 Ri + Rj + Rk ⊂ H. Gibbs didn't consider the quaternion product to be a \fundamental notion" in vector analysis though, and argued for a vector analysis that would apply in arbitrary dimension. Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras This rivalry between notations flared into a war in the late 1800's between the `quaternionists' and the `vectorists'. Willard Gibbs (1839-1903) Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras The most prominent quaternionist was James Clerk Maxwell. \The invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can only be compared, for its importance, with the invention of triple coordinates by Descartes. The ideas of this calculus, as distinguished from its operations and symbols, are fitted to be of the greatest use in all parts of science." James Clerk Maxwell (1831-1879) Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras There was also Hamilton's student Peter Tait: \Even Prof. Willard Gibbs must be ranked as one the retarders of quaternions progress, in virtue of his pamphlet on Vector Analysis, a sort of hermaphrodite monster, compounded of the notation of Hamilton and Grassman." Peter Tait (1831-1901) Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras On the vectorist side was Lord Kelvin: \Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell." William Thomson (Lord Kelvin, 1824-1907) Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras Ultimately, the superiority and generality of vector notation won out. Certain useful fragments of Hamilton's quaternions do remain in standard calculus courses however. For example, we still use the \right-hand rule" i × j = k in multivariable calculus. Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras Let's write (−1; −1; R) in place of H. This notation suggests a number of ways to generalize H. For instance, let (1; 1; R) be the R-algebra with basis f1; i; j; ijg and defining relations i 2 = 1 j2 = 1 ij = −ji: 0 1 1 0 Then (1; 1; ) =∼ M ( ) via i; j 7! ; . R 2 R 1 0 0 −1 0 −1 1 0 Also, (1; −1; ) =∼ M ( ) via i; j 7! ; . R 2 R 1 0 0 −1 Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras More generally, we can define (a; b; R) to be the R-algebra with basis f1; i; j; ijg and defining relations 2 2 ∗ i = a j = b ij = −ji a; b 2 R : ∼ 2 2 × Because (a; b; R) = (ax ; by ; R) for all x; y 2 R , we conclude that ∼ (a; b; R) = H if a; b < 0 and ∼ (a; b; R) = M2(R) otherwise. Thus (a; b; R) is either a division algebra or else M2(R). Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras There are other ways that we could have generalized (−1; −1; R). If F is a field of characteristic zero and a; b 2 F ∗ we can define the generalized quaternion algebra (a; b; F ). Let F = Q and consider the Q-algebra (−1; −1; Q). As (−1; −1; Q) ( (−1; −1; R), we see that (−1; −1; Q) is a division algebra. ∼ As before we also see that (1; 1; Q) = M2(Q). Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras Recall that if F = R,(a; b; F ) is a division algebra or else M2(R). Theorem (Wedderburn) For any field F , if the F -algebra (a; b; F ) is not a division algebra then ∼ (a; b; F ) = M2(F ). Joseph Wedderburn (1882-1948) Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras Note that the case F = R is special in that in general there will not be a unique quaternion division algebra over F . Theorem (Classification of quaternion algebras over Q) Let A = (a; b; Q) be a quaternion algebra over Q. Then the set of primes p such that (a; b; Qp) is a division algebra over Qp is finite with an even number of elements. Conversely, for any finite set S of primes with even cardinality there exists a unique (up to isomorphism) quaternion algebra (a; b; Q) such that (a; b; Qp) precisely when p 2 S. Consequence: There are infinitely many quaternion division algebras over Q. Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras A \complex" example: F = C. Let A be a quaternion algebra over C. By the fundamental theorem of algebra, A cannot be a division algebra. ∼ Then by Wedderburn's theorem, A = M2(C). Thus M2(C) is the only quaternion algebra over C. Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras Over the finite field Fq : × Let D be a quaternion division algebra over Fq and let D act on D by conjugation. When we count orbits with the class equation we get X X q4 − 1 jD×j = q4 − 1 = q − 1 + [D× : C (d)×] = q − 1 + : D q2 − 1 d d Therefore there exists m 2 Z such that q4 − 1 = q − 1 + m(q2 + 1); a contradiction. Thus M2(Fq) is the only quaternion algebra over Fq. Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras The reduced norm: Suppose that A = (a; b; F ) is a quaternion algebra over a field F with char(F ) 6= 2. Given a quaternion α = t + xi + yj + zk we define an involution on A by α = t − xi − yj − zk: Note that αα = t2 − ax2 − by 2 + abz2. When A = (−1; −1; F ) this simplifies to the pleasing αα = t2 + x2 + y 2 + z2: Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras The reduced norm of α 2 A is defined to be nr(α) = αα. Note that nr(·) is multiplicative. If α; β 2 A then nr(αβ) = αβ = (αβ)(αβ) = αββα = αnr(β)α = ααnr(β) = nr(α)nr(β): Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras An application of the reduced norm: Theorem (Euler, 1749) 2 2 2 2 2 2 2 2 (a1 + a2 + a3 + a4)(b1 + b2 + b3 + b4) = 2 (a1b1 − a2b2 − a3b3 − a4b4) + 2 (a1b2 + a2b1 + a3b4 − a4b3) + 2 (a1b3 − a2b4 + a3b1 + a4b2) + 2 (a1b4 + a2b3 − a3b2 + a4b1) Leonhard Euler (1707-1783) Euler's \four squares identity"' shows that the product of two sums of four squares is again a sum of four squares. Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras An easy proof of Euler's four squares identity: Let α; β 2 (−1; −1; R) be given by α = a1 + a2i + a3j + a4k; and β = b1 + b2i + b3j + b4k: 2 2 2 2 2 2 2 2 Then nr(α) = a1 + a2 + a3 + a4 and nr(β) = b1 + b2 + b3 + b4.