QUATERNIONS AND THEIR GENERALIZATIONS

Leonard Eugene Dickson

Read before the Academy, April 26, 1921 Proc. Nat. Acad. Sci. U.S.A. 7 (1921): 109-114.

1. The discovery of quaternions by W. R. Hamilton in 1843 has led to an extensive theory of linear algebras (or closed systems of hypercomplex numbers) in which the quaternion algebra plays an important rôle. Frobenius^{1} proved that the only real linear associative algebras in which a product is zero only when one factor is zero are the real number system, the ordinary complex number system, and the algebra of real quatermons. A much simpler proof has been given by the writer.^{2} Later, the writer^{3} was led to quaternions very naturally by means of the four-parameter continuous group which leaves unaltered each line of a set of rulings on the quadric surface x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} = 0.

The object of the present note is to derive the algebra of quaternions and its direct generalizations without assuming the associative or commutative law. I shall obtain this interesting result by two distinct methods.

2. The term field will be employed here to designate any set of ordinary complex numbers which is closed under addition, subtraction, multiplication, and division. Thus all complex numbers form a field, likewise all real numbers, or all rational numbers.

Just as a couple (a, b) of real numbers defines an ordinary complex number a + bi, where i^{2} = −1, so also an n-tuple (x_{1}, ..., x_{n}) of numbers of a field F defines a hypercomplex number

x = x_{1}e_{1} + x_{2}e_{2} + ... + x_{n}e_{n},  (1)

where the units e_{1}, ..., e_{n} are linearly independent with respect to the field F and possess a multiplication table

e_{i}e_{j} = ∑_{k=1}^{n} γ_{ijk}e_{k}  (i, j = 1, ..., n),  (2)

1Frobenius, Jour. für Math., 84, 1878 (59).

2Dickson, Linear Algebras, “Cambridge Tracts in Mathematics and Mathematical Physics,”

No. 16, 1914 (10-12).

3Dickson, Bull. Amer. Math. Soc., 22, 1915 (53-61).

in which the γ’s are numbers of F. Let x′ = ∑ x_{i}e_{i} be another hypercomplex number whose coördinates x′_{i} are numbers of F. Then shall

xx′ = ∑_{i,j=1}^{n} x_{i}x′_{j}.e_{i}e_{j}, x ± x′ = ∑_{i=1}^{n}(x_{i} ± x′_{i})e_{i}, fx = xf = ∑_{i=1}^{n} fx_{i}.e_{i},

when f is in F, so that multiplication is distributive. Under these assumptions, the set of all numbers (1) with coördinates in F shall be called a linear algebra over F.

3. We assume that e_{1} is a principal unit (modulus), so that e_{1}x = xe_{1} = x for every number x of the algebra, and write 1 for e_{1}. We assume that every number of the algebra satisfies a quadratic equation with coefficients in F. If e^{2} + 2ae + b = 0, (e + a)^{2} = a^{2} − b, so that we may take the units to be 1, E_{2}, ..., E_{n}, where E_{i}^{2} = s_{ii}, a number of F. For i and j distinct and > 1, E_{i} ± E_{j} satisfies a quadratic, so that (E_{i} ± E_{j})^{2} = s_{ii} + s_{jj} ± (E_{i}E_{j} + E_{j}E_{i}) is a linear function of E_{i} ± E_{j}. Thus E_{i}E_{j} + E_{j}E_{i} is a linear function of E_{i} + E_{j} and of E_{i} − E_{j} and hence is a number 2s_{ij} = 2s_{ji} of F.

Let u_{2}, ..., u_{n} be arbitrary numbers of F and write U = ∑ u_{k}E_{k}. Then U^{2} equals

Q = ∑_{k,l=2}^{n} s_{kl}u_{k}n_{l}.

It is a standard theorem that Q can be reduced to ∑ c_{i}v_{i}^{2} by a linear transformation u_{k} = ∑ a_{kl}v_{l} with coefficients in F and of determinant ≠ 0. Write

e_{l} = ∑_{k,l=2}^{n} a_{kl}E_{k} (l = 2, ..., n).

Then 1, e_{2}, ..., e_{n} are linearly independent and may be taken as the new units of our algebra over F. Then

\begin{aligned}U & = ∑_{k,l} a_{kl}v_{l}E_{k} = ∑_{l} v_{l}e_{l}, \\ U^{2} & = ∑_{k,l} v_{k}v_{l}e_{k}e_{l} = ∑ c_{i}v_{i}^{2}. \end{aligned}

Hence

e_{i}^{2} = c_{i},  e_{i}e_{j} + e_{j}e_{i} = 0  (i, j = 2, ..., n, i ≠ j).  (3)

4. Write x = x_{1} + ∑ x_{i}e_{i}. Then (x − x_{1})^{2} = ∑ c_{i}x_{i}^{2}. This gives xx′ = x′x = σ, where

x′ = 2x_{1} − x = x_{1} − ∑_{i=2}^{n} x_{i}e_{i}, σ = x_{1}^{2} − ∑_{i=2}^{n} c_{i}x_{i}^{2}.

We shall call x′ the conjugate to x and σ = σ(x) the norm of x. Hence the product of any number and its conjugate in either order equals its norm. We assume that the norm of a product equals the product of the norms of the factors:

σ(x)σ(ξ) = σ(X),  if xξ = X,  (4)

and shall investigate the resulting types of linear algebras. We assume also that each c_{i} ≠ 0 in (3).

5. By (2) the coördinates of X = xξ are X_{k} = ∑ x_{i}ξ_{j}γ_{ijk}. Since e_{i}^{2} = c_{i}, we have γ_{ii1} = c_{i}, γ_{iik} = 0 (i > 1, k > 1). Hence

\begin{aligned}X_{1} & = x_{1}ξ_{1} + ∑_{i=2}^{n} x_{i}ξ_{i}c_{i} + ∑_{\begin{matrix}i,j=2, \\ i≠j \end{matrix}}^{n} x_{i}ξ_{j}γ_{ij1}, \\ X_{k} & = x_{1}ξ_{k} + x_{k}ξ_{1} + ∑_{\begin{matrix}i,j=2 \\ i≠j \end{matrix}}^{n} x_{i}ξ_{j}γ_{ijk}  (k > 1). \end{aligned}  (5)

Since this transformation is the identity X = x if ξ = 1, we obtain an infinitesimal transformation by taking ξ_{1} = 1, ξ_{j} = δt, ξ_{i} = 0 (i ≠ 1, j):

\begin{aligned}δx_{1} = X_{1} − x_{1} & = \{c_{j}x_{j} + ∑_{\begin{matrix}i=2 \\ i≠j \end{matrix}}^{n} γ_{ij1}x_{i}\} δt, \\ δx_{j} & = \{x_{1} + ∑_{\begin{matrix}i=2 \\ i≠j \end{matrix}}^{n} y_{ijj}x_{i}\} δt, \\ δx_{k} & = \{∑_{\begin{matrix}i=2 \\ i≠j \end{matrix}}^{n} γ_{ijk}x_{i}\} δt  (k ≠ 1, j). \end{aligned}  (6)

For these ξ’s, σ(ξ) is unity to within an infinitesimal of the second order. Hence the increment to σ(x) must vanish identically, so that

γ_{ij1} = γ_{ijj} = γ_{iji} = 0  (1, i, j distinct),  (7)
c_{k}γ_{ijk} + c_{i}γ_{kji} = 0  (1, i, j, k distinct).  (8)

By (7), (5) simplifies to

X_{1} = x_{1}ξ_{1} + ∑_{i=2}^{n} x_{i}ξ_{i}c_{i}, X_{k} = x_{1}ξ_{k} + x_{k}ξ_{1} + ∑ x_{i}ξ_{j}γ_{ijk} (k > 1), (9)

where, in the final sum, i and j range over distinct values from 2, ..., n excluding k. This final sum is, therefore, absent if^{4} n = 3; whence σ(X) has the term 2x_{2}ξ_{2}c_{2}.x_{3}ξ_{3}c_{3} which does not occur in σ(x)σ(ξ). But c_{2}c_{3} ≠ 0 by hypothesis. Hence n > 3.

4By (7), e_{2}e_{3} = e_{3}e_{2} = 0.

Hitherto we have not examined the conditions which follow from the final equations (3); these are

γ_{jik} = −γ_{ijk}  (i, j = 2, ..., n; i ≠ j).  (10)

6. Taking n = 4 and applying (10), we see that (9) become

\begin{cases}X_{1} & = x_{1}ξ_{1} + c_{2}x_{2}ξ_{2} + c_{3}x_{3}ξ_{3} + c_{4}x_{4}ξ_{4}, \\ X_{2} & = x_{1}ξ_{2} + x_{2}ξ_{1} + γ_{342}(x_{3}ξ_{4} − x_{4}ξ_{3}), \\ X_{3} & = x_{1}ξ_{3} + x_{3}ξ_{1} + γ_{243}(x_{1}ξ_{4} − x_{4}ξ_{1}), \\ X_{4} & = x_{1}ξ_{4} + x_{4}ξ_{1} + γ_{234}(x_{2}ξ_{3} − x_{3}ξ_{2}). \end{cases}  (11)

These transformations do not in general form a group and hence are not generated by the corresponding infinitesimal transformations employed above. Hence it remains to require that σ(X) = σ(x)σ(ξ) under the transformations (11). The conditions are seen to be

c_{3}c_{4} = −c_{2}γ_{342}^{2}, c_{2}c_{4} = −c_{3}γ_{243}^{2}, c_{2}c_{3} = −c_{4}γ_{234}^{2}, c_{4}γ_{234} = c_{2}γ_{342} = −c_{3}γ_{243},

the first two of which reduce to the third by means of the last three equations. To these last can be reduced all the conditions (8) by means of (10).

Applying the transformation of variables which multiplies x_{4}, ξ_{4}, X_{4} by γ_{234}, and leaves the remaining x_{i}, ξ_{i}, X_{i} unaltered, we get

\begin{cases}X_{1} & = x_{1}ξ_{1} + c_{2}x_{2}ξ_{2} + c_{3}x_{3}ξ_{3} − c_{2}c_{3}x_{4}ξ_{4}, \\ X_{2} & = x_{1}ξ_{2} + x_{2}ξ_{1} − c_{3}x_{3}ξ_{4} + c_{3}x_{4}ξ_{3}, \\ X_{3} & = x_{1}ξ_{3} + x_{3}ξ_{1} + c_{2}x_{2}ξ_{4} − c_{2}x_{4}ξ_{2}, \\ X_{4} & = x_{1}ξ_{4} + x_{4}ξ_{1} + x_{2}ξ_{3} − x_{3}ξ_{2}. \end{cases}  (11′)

These are the values obtained by Lagrange^{5} in his generalization σ(x)σ(ξ) = σ(X) of Euler’s formula for the product of two sums of four squares. Then xξ = X gives the following multiplication table for the units:

\{ \begin{matrix}e_{2}^{2} = c_{2} e_{3}^{2} = c_{3}, e_{4}^{2} = −c_{2}c_{3}, e_{2}e_{3} = e_{4}, e_{3}e_{2} = −e_{4}, \\ e_{2}e_{4} = c_{2}e_{3}, e_{4}e_{2} = −c_{2}e_{3}, e_{3}e_{4} = −c_{3}e_{2}, e_{4}e_{3} = c_{3}e_{2}. \end{matrix}  (12)

This algebra is associative and is the direct generalization of quaternions to a general field F which the writer^{6} obtained elsewhere from assumptions including associativity. The four-rowed determinants of the general number x of this algebra equals σ^{2}(x). The case c_{2} = c_{3} = −1 gives the algebra of quaternions, for which it is customary to write i, j, k instead of our units e_{2}, e_{3}, e_{4}.

5Lagrange, Nouv. Mém. Acad. Roy. Sc. de Berlin, année 1770, Berlin, 1772 (123-133); Oeuvres de Lagrange, 3, 1869 (189). Reproduced in Dickson’s History of the Theory of Numbers, II, 1920 (279-281).

6Dickson, Trans. Amer. Math. Soc., 13, 1912 (65).

7. It is not very laborious to show by the above method that the cases n = 5 and n = 6 are excluded. However, Hurwitz has proved that a relation of the form σ(x)σ(ξ) = σ(X) is impossible if n ≠ 1, 2, 4, 8. A slight simplification of his proof, together with an account of the history of this problem, has been given by the writer.^{7} Hurwitz made no attempt to find all solutions when n = 4. We proceed to treat this problem.

Consider the case c_{i} = −1 to which the general case may be reduced by an irrational transformation. Then σ(x) = ∑ x_{i}^{2}. We investigate the linear algebras having property (4), i. e.,

(x_{1}^{2} + ... + x_{n}^{2})(ξ_{1}^{2} + ... + ξ_{n}^{2}) = X_{1}^{2} + ... + X_{n}^{2},  (13)

if

X_{k} = ∑_{j=1}^{n} (∑_{i=1}^{n} γ_{ijk}x_{i}) ξ_{j}  (k = 1,..., n).  (14)

The matrix M of this substitution has the element ∑_{i=1}^{i=n} γ_{ijk}x_{i} in the kth row and jth column. If this substitution is applied to a quadratic form in X_{1}, ..., X_{n} of matrix Q, it is a standard theorem that we obtain a quadratic form in ξ_{1}, ..., ξ_{n}, whose matrix is M′QM, where M′ is the transposed of M, being obtained from M by the interchange of its rows and columns. In our problem, Q is the identity matrix I whose elements are all zero except the diagonal elements which are 1. Hence, by (13),

M′M = (x_{1}^{2} + ... + x_{n}^{2})I.  (15)

When a homogeneous polynomial σ(x_{1}, ..., x_{n}) of any degree has the property (4) of possessing a theorem of multiplication, the writer^{8} has proved that we may apply a linear transformation on x_{1}, ..., x_{n} which leave σ(x) unaltered and one on ξ_{1}, ..., ξ_{n} which leaves σ(ξ) unaltered such that the new algebra has the principal unit e_{1}, so that γ_{1jk} and γ_{j1k} are both 0 if j ≠ k, and both unity if j = k.

Hence M = x_{1}M_{1} + ... + x_{n}M_{n}, where γ_{ijk} is the element in the kth row and jth column of M_{i}, whence M_{i} = I. Thus (15) gives

M′_{i} = −M_{i}, M′_{i}M_{i} = I, M′_{i}M_{j} + M′_{j}M_{i} = 0 (i > 1, j > 1, j ≠ i). (16)

7Dickson, Annals of Math., 20, 1919 (155-171, 297).

8Dickson, Comptes Rendus du Congrès Internat. Math., Strasbourg, 1920 (131-146).

In view of the values of γ_{jik}, and M′_{i} = −M_{i}, we have, when n = 4,

\begin{aligned}M_{2} & = (\begin{matrix}0 & −1 & 0 & 0 \\ 1 & 0 & −γ_{223} & −γ_{224} \\ 0 & γ_{223} & 0 & −γ_{234} \\ 0 & γ_{224} & γ_{234} & 0 \end{matrix}) , \\ M_{3} & = (\begin{matrix}0 & 0 & −1 & 0 \\ 0 & 0 & −γ_{323} & −γ_{324} \\ 1 & γ_{323} & 0 & −γ_{334} \\ 0 & γ_{324} & γ_{334} & 0 \end{matrix}) , \\ M_{3} & = (\begin{matrix}0 & 0 & 0 & −1 \\ 0 & 0 & −γ_{423} & −γ_{424} \\ 0 & γ_{423} & 0 & −γ_{434} \\ 1 & γ_{424} & γ_{434} & 0 \end{matrix}) . \end{aligned}

By M′_{i}M_{i} = I, we have M^{2^{i}} = −I, which gives

γ_{223} = γ_{224} = γ_{323} = γ_{334} = γ_{424} = γ_{434} = 0, γ^{2} = δ^{2} = ϵ^{2} = 1,

where γ = γ_{234}, δ = γ_{324}, ϵ = γ_{423}. Hence

\begin{aligned}M_{2} & = (\begin{matrix}0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −γ \\ 0 & 0 & γ & 0 \end{matrix}) , \\ M_{3} & = (\begin{matrix}0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −δ \\ 1 & 0 & 0 & 0 \\ 0 & δ & 0 & 0 \end{matrix}) , \\ M_{4} & = (\begin{matrix}0 & 0 & 0 & −1 \\ 0 & 0 & −ϵ & 0 \\ 0 & ϵ & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}) . \end{aligned}  (17)

The final condition (16) states that M_{i}M_{j} is skew-symmetric. The products M_{2}M_{3}, M_{2}M_{4}, M_{3}M_{4} of matrices (17) are seen at once to be skew-symmetric if and only if δ = −γ, ϵ = γ, and then M_{2}M_{3} = γM_{4}, M_{2}M_{4} = −γM_{3}, M_{3}M_{4} = γM_{2}. Writing i, j, k for M_{2}, M_{3}, γM_{4}, we have the multiplication table of quaternions. Or we may form the matrix M and write X_{k} for the sum of the products of the elements of its kth row by ξ_{1}, ..., ξ_{4}, and take γ = 1 (by multiplying x_{4}, ξ_{4}, X_{4} by γ); we obtain (11′) for c_{2} = c_{3} = −1. Hence we have again obtained the quaternion algebra without assuming the associative law. The case n = 8 is being investigated in this way by one of my students.