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Home , Division algebra, Quaternion

An introduction to

Benjamin Linowitz

Department of Oberlin College An Introduction to Quaternion Algebras

One of the many nice features of the complex is that their can be described using the geometric operations of stretching and .

The importance of the complex numbers in two-dimensional was recognized very early.

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

William Rowan Hamilton sought a 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 .

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 ∈ 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 ∈ R.

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

Hamilton did, however, find a four dimensional real .

Theorem (Hamilton, 1843) The R-algebra H with {1, i, j, ij} and defining relations

i 2 = −1 j2 = −1 ij = −ji

is a four-dimensional division algebra. (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 was pioneered by Willard Gibbs.

In his 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 ∈ Ri + Rj + Rk ⊂ H.

Gibbs didn’t consider the quaternion to be a “fundamental notion” in vector analysis though, and argued for a vector analysis that would apply in arbitrary .

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 .

“The invention of the calculus of is a step towards the knowledge of quantities related to 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 :

“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 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 {1, i, j, ij} 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 {1, i, j, ij} and defining relations

2 2 ∗ i = a j = b ij = −ji a, b ∈ R .

∼ 2 2 × Because (a, b, R) = (ax , by , R) for all x, y ∈ 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 ∈ F ∗ we can define the generalized (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 ) quaternion algebra (a, b, Q) such that (a, b, Qp) precisely when p ∈ 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 |D×| = q4 − 1 = q − 1 + [D× : C (d)×] = q − 1 + . D q2 − 1 d d

Therefore there exists m ∈ 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 :

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 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 ∈ A is defined to be nr(α) = αα.

Note that nr(·) is multiplicative. If α, β ∈ 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 ∈ (−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.

The identity now follows from the fact that nr(α)nr(β) = nr(αβ) and the fact that in (−1, −1, R) the reduced norm of an element (i.e., αβ) is always the sum of four squares.

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

Definition: An of an algebra over Q is a which is also a finitely generated Z-module containing an Q-basis of the algebra.

Example 1: Z[i] is a quadratic order of the Q-algebra Q(i).

Example 2: M2(Z) is a maximal order of M2(Q).

Example 3: The subring

Z[i, j] = Z + Zi + Zj + Zk of the Q-algebra (a, b, Q) is an order provided that a, b ∈ Z.

Remark: If F is a number field with of OF then all of this generalizes to F -algebras by replacing Q with F and Z with OF . Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

An application to

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

Theorem (Lagrange, 1770) Every can be expressed as the sum of four squares.

Joseph-Louis Lagrange (1736-1813)

We’ll use quaternion orders to give a short proof of this theorem.

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

Let A = (−1, −1, Q).

We’ve already seen that Z[i, j] = Z + Zi + Zj + Zk is an order of A.

Z[i, j] is not a maximal order of A.

√ √ This is analogous to the fact that Z[ −3] is an order of Q( −3) but is not√ the maximal order (i.e., ring of integers) since this is Z[(−1 + 3)/2].

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

√ √ Our analogy with Z[ −3] and Z[(−1 + 3)/2] is not an accident.

Indeed, the element α = i + j + k satisfies α2 + 3 = 0, so it is natural to define −1 + i + j + k ω = , 2 which also satisfies ω2 + ω + 1 = 0.

It turns out that the order

O = Z + Zi + Zj + Zω = Z[i, j] + Z[i, j]ω

is the unique maximal order of A which contains Z[i, j].

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

Unit groups:

× ∼ Z[i, j] = {±1, ±i, ±j, ±k} = Q8, the quaternion of order 8.

× S O = Q8 (±1 ± i ± j ± k)/2, is a group of order 24 isomorphic to SL2(F3).

Easy lemma 1: For all β ∈ O there exists γ ∈ O× such that βγ ∈ Z[i, j].

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

Lemma 2: Let p be a prime. There exists π ∈ O such that nr(π) = p.

Proof: If p = 2 then we have nr(1 + i) = 12 + 12 = 2. So suppose p is odd.

∼ ∼ Then O/pO = (−1, −1, Fp) = M2(Fp) and there exists a left I mod p ⊂ O/pO with dimFp (I mod p) = 2.

The preimage of I in O must be principal, hence there exists β ∈ (−1, −1, Q) such I = βO.

There’s a theory of norms for ideals of orders, which says that N(I ) = |O/I | = p2 and that N(βO) = nr(β)2.

Therefore nr(β) = p.

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

We now prove Lagrange’s theorem. Let n be a natural number.

It suffices to find an element of α ∈ Z[i, j] with nr(α) = n.

By multiplicativity of the reduced norm it suffices to consider the case n = p is prime.

By Lemma 2 there exists β ∈ O such that nr(β) = p.

× By Lemma 1 there exists γ ∈ O such that α := βγ ∈ Z[i, j].

× × But nr(γ) ∈ Z , and because Z = {±1} and nr(γ) must be a sum of four squares and hence positive, we conclude that nr(γ) = 1 and nr(α) = p.

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

Geometric applications

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

Theorem (Siegel, 1944)

Given a positive d, let hd , Rd denote the class number and regulator of the unique quadratic order of discriminant d. Then

X π2 h R = X 3/2 + O(X log X ). d d 18ζ(3) d

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

On the other hand, if one counts real quadratic fields by regulator instead of discriminant then one can obtain separation.

Theorem (Sarnak, 1982) As X → ∞ we have

X e2X h ∼ . d 2X Rd

Something that makes Sarnak’s proof especially novel is that the main ideas arise from differential geometry.

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

(Rough!) Sketch of Proof

Consider the modular surface S = H/SL2(Z).

The lengths of closed geodesics on S correspond to regulators of real quadratic fields.

These lengths appear with multiplicity equal to the class number.

The Prime Geodesic Theorem in this context gives an asymptotic formula for the number of closed geodesics with length less than X .

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

Moral: The lengths of geodesics of certain hyperbolic surfaces encode interesting number theoretic information.

Given a Riemannian manifold M, let LS(M) denote the length spectrum of M; that is, LS(M) is the set of lengths of closed geodesics on M.

The following information is all determined by LS(M): dim(M) curvature volume(M) arithmeticity

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

Let A = (a, b, Q) be a quaternion algebra over Q with at least one of a, b positive.

Let O be a maximal order of A.

∼ Then O ⊂ (a, b, Q) ⊂ (a, b, R) = M2(R).

Let Γ(1) denote the subgroup of SL2(R) consisting of those elements of O with reduced norm 1 (equiv., 1).

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

As a subgroup of SL2(R), the group Γ(1) acts on the complex upper half H by fractional linear transformations: az + b z 7→ . cz + d The resulting space X (1) = H/Γ(1) is a Shimura .

∼ X (1) is a hyperbolic surface which is compact when A =6 M2(Q).

∼ When A = M2(Q) the resulting Shimura curve is simply the modular surface.

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

Theorem (Vign´eras, 1980) There exist Shimura X (1) and Y (1) which are not isomorphic yet satisfy

LS(X (1)) = LS(Y (1)).

√ 1. Vign´eras’Shimura curves are defined over Q( 10) and arise from the same quaternion algebra.

2. They are also isospectral with respect to the Laplace operator (by the Selberg Trace Formula).

Benjamin Linowitz An introduction to quaternion algebras An Introduction to Quaternion Algebras

Geometrically, Shimura curves arising from the same quaternion algebra are commensurable.

Theorem (Reid, 1992) If X (1) and Y (1) are Shimura curves satisfying

LS(X (1)) = LS(Y (1)) then X (1) and Y (1) are commensurable and arise from the same quaternion algebra.

Idea of proof: Lengths of geodesics on Shimura curves arise from maximal subfields of the associated quaternion algebra, and a quaternion algebra over a number field is determined by its maximal subfields.

Benjamin Linowitz An introduction to quaternion algebras