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QUATERNIONS AND THEIR GENERALIZATIONS
Department of MathematICS, UNIVERSITY OF CHICAGO
Read before the Academy, April 26, 1921
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' 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 quaternions. A much simpler proof has been given by the writer. Later, the writer3 was led to quaternions very naturally by means of the fourparameter continuous group which leaves unaltered each line of a set of rulings on the quadric surface x + x; + x; + x; = 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 i2 = 1, so also an n-tuple (x1,.., x) of numbers of a field F defines a hypercomplex number
where the units e1, ..., e, are linearly independent with respect to the field F and possess a multiplaction table
n which the y's are numbers of F. Let x' = Exe; be another hypercomplex number whose coördinates x; are numbers of F.
when ƒ 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, is a principal unit (modulus), so that e1x = xe = 1 for every number x of the algebra, and write 1 for e1. We assume that every number of the algebra satisfies a quadratic equation with coefficients in F. If e2 + 2ae + b ·0, (e + a)2 a2-b, so that we may take the units to be 1, E2, ..., En, where E = Sii, a number of F.
For i and
j distinct and >1, E¡ ± E; satisfies a quadratic, so that (E; ± E;)2 Sii + S;; ± (E¡E; + E¡E;) is a linear function of E; ± E;. Thus E¡E; + E,E, is a linear function of E; + E, and of E;-E;, and hence is a number 25; 25;; of F.
un be arbitrary numbers of F and write U
It is a standard theorem that Q can be reduced to Σc,v,2 by a linear transformation uk = Zak with coefficients in F and of determinant 0.
Then 1, €2,
k,l = 2
..., en are linearly independent and may be taken as the new units of our algebra over F. Then
We shall call x' the conjugate to x and σ = o(x) the norm of x.
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), if x = X,
and shall investigate the resulting types of linear algebras. We assume also that each c; ‡ 0 in (3).
5. By (2) the coördinates of X = xέ are X we have Yil = Cj, Yiik 0 (i>1, k>1). Hence
= 2xYjk. Since e
For these 's, σ(§) is unity to within an infinitesimal of the second order. Hence the increment to σ(x) must vanish identically, so that
where, in the final sum, i and j range over distinct values from 2, excluding k. This final sum is, therefore, absent if n=3; whence (X)
has the term 2x2§2C2.X3§3C3 which does not occur in σ(x)o(§). But C2C3 ‡0 by hypothesis. Hence n>3.
Hitherto we have not examined the conditions which follow from the final equations (3); these are
6. Taking n = 4 and applying (10), we see that (9) become
( X1 = X1§1 + €2X2§2 + €3X3§3 + €4X4§4, X2=X1§2 + X2§1 + Y342(X3§4−X4§3), (11) X3=X1§3+ X3§1 + Y243(X2§4 — X4§1), X4=X1§4 + X4§: + √234(X2§3—X3§2). 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) (x)o() under the transformations (11). The conditions are seen to be
C3C1 = -C2Y 342, C2C4 = C3Y 243, C2C3 = -C47 234, C4Y234 = C2Y342 = -C3Y243, 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 x4, 4, X4 by 1234, and leaves the remaining x,, ;, X, unaltered, we get JX1=X1§1 + €2X2§2 + C3X3§3 — C2C3X4§4, X2=X1§2 + X2§1 −C3X3§4 + C3X4§3, X3=X1§3 + X3§1 + C2X2§4−C2X4$2, X4=X1§4 + X4§1 + X2§3−X3§2. These are the values obtained by Lagrange in his generalization σ (x)o(§) (X) of Euler's formula for the product of two sums of four squares. Then x X gives the following multiplication table for the units:
This algebra is associative and is the direct generalization of quaternions to a general field F which the writer obtained elsewhere from assumptions including associativity. The four-rowed determinants of the general number x of this algebra equals o2(x). The case c2 = C3 = 1 gives the algebra of quaternions, for which it is customary to write i, j, k instead of our units €2, C3, €4.
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 o(x)o() = (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. Hurwitz made no attempt to find all solutions when n = 4. We proceed to treat this problem. -1 to which the general case may be reduced by an irrational transformation. Then σ(x) = Ex. We investigate the linear algebras having property (4), i. e.,
Consider the case c;
The matrix M of this substitution has the element i=1; in the kth row and jth column. If this substitution is applied to a quadratic form in X1,...,X, of matrix Q, it is a standard theorem that we obtain a quadratic form in §1, ..., En, 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),
When a homogeneous polynomial σ(x1, ..., x) of any degree has the property (4) of possessing a theorem of multiplication, the writers has proved that we may apply a linear transformation on x1, Xn which leave o(x) unaltered and one on §1, ..., En which leaves σ(§) unaltered such that the new algebra has the principal unit e1, so that Yijk and Yjlk are both 0 if jk, and both unity if j
x1M1 + +xnMn where
Hence M Yijk kth row and jth column of M;, whence M; = I.
is the element in the Thus (15) gives
M = −M;, M;M; = 1, M';M; + M';M; = 0 (i>1, j>1, j + i). (16)
In view of the values of Yjik, and M
-M1, we have, when n
Y223 O -Y234
0 Y224 234 0
Y324 Y334 0
The final condition (16) states that M¡M; is skew-symmetric. products M2M3, M2M4, M3M4 of matrices (17) are seen at once to be