Quasi-associativity and Cayley-Dickson algebras

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . It is shown that, if the imaginary part of a hypercomplex number in more than 2 dimensions is squared to be non-positive, the linearly independent imaginary units defining the hypercomplex number should anti-commute with each other. The simplest examples are Hamilton’s quaternions (4 dimensions) and octonions (8 dimensions). This leads to an inductive construction of Cayley–Dickson algebras A2n via “quasi-associativity” for positive integer n ≥ 3. The “quasiassociativity” guaranteeing anti-commutativity among the imaginary units for non-associative algebras is a mixture of associativity and non-associativity and can be formulated inductively. It allows us to compute Cayley–Dickson products to define A2n for n > 3 that are no longer alternative but flexible. In other words, general Cayley–Dickson algebras A2n for n ≥ 3 are defined by the “quasi-associativity” for the product of basis elements. In addition, a new 2D quaternion-valued matrix representation of quaternions, which is equivalent to the well known Pauli representation, is also proposed. The Study determinant of the quaternionic matrix, which obeys the composition law, is modified such that the composition law of quaternions is derived from the composition property of a modified Study (Study-like) determinant. It can easily be generalized to arbitrary Cayley–Dickson algebras A2n , although the associativity of the matrix product is lost for n ≥ 3. In addition, the associativity of the product of the matrix elements is to be replaced by the “quasi-associativity” for n ≥ 3. It turns out that the composition property of the Study-like determinant holds true only for quaternions and octonions in accord with the Hurwitz theorem. The same is true, of course, for real and complex numbers that are also represented by 2 × 2 matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


Introduction
The physical applications of division, Jordan, and related algebras to particle physics have been attempted by many authors in the past [1].This note treats, from a physicist's point of view, some mathematical aspects of Cayley-Dickson algebras A 2 n over the real field, which clarify a possible connection of hypercomplex numbers in 2 n dimensions for positive integer n with a 2 × 2 matrix, although the associativity of the matrix product no longer holds true for n ≥ 3.
It is well known that A 1 = R, A 2 = C, A 4 = H, and A 8 = O are, respectively, real, complex, quaternion fields and octonions.It is also well known that quaternions have only one irreducible representation that is 2D, namely, the Pauli representation.On the other hand, since O is a nonassociative algebra, it has no matrix representation.In this note, we propose a new 2D representation of quaternions, which is equivalent to the Pauli representation, and generalize it to octonions and more general Cayley-Dickson algebras.Care must be taken, however, since, for n > 2, the algebra A 2 n is not associative so that the corresponding matrix representation no longer obeys the associative law.Nonetheless, the (non-associative) representation is homomorphic and the Study determinant [2] PTEP 2014, 013A03 K. Morita for a special class of quaternion-valued matrices in the new representation can be modified in a way generalizable to the case n > 2. The modified Study determinant is called a Study-like determinant.
To this purpose we construct higher-dimensional hypercomplex numbers than quaternions by induction, which requires the "quasi-associativity" for the product of basis elements of A 2 n for n ≥ 3. The quasi-associativity, which is a mixture of associativity and non-associativity, naturally leads to the Cayley-Dickson process.Then, it turns out that the new 2 × 2 matrix representation of quaternions is straightforwardly generalized to octonions, sedenions, and more general Cayley-Dickson algebras.
We shall first prove in the next section that Hamilton's triplet with only two imaginary units {i, j} in addition to the unit element 1 is not allowed as an associative division algebra closed under multiplication.Although this has been very well known for a long time, our proof, the last stage of which is based on May's classroom note [3], strongly suggests a consistent introduction of one more imaginary unit k without recourse to the composition law.In Sect.3, we thereby derive Hamilton's quaternions from the assumption that the linear combination of three independent imaginary units {i, j, k} with real coefficients is squared to be non-positive [4].In fact, we prove Hamilton's formula i jk = −1 from this assumption only provided that associativity and divisibility are taken for granted.We also introduce a new matrix representation equivalent to the Pauli representation and argue that the Study-like determinant satisfies the composition law like the Study determinant does, proving the composition law for quaternions.
In Sect.4, octonions are introduced by defining the "quasi-associative law" among the imaginary units {i, j, k, li, l j, lk, l} where another imaginary unit l anti-commuting with {i, j, k} is assumed to exist. 1 The "quasi-associativity" is introduced to avoid the Frobenius theorem in algebra, which denies the existence of such an extra imaginary unit anti-commuting with Hamilton's units {i, j, k} unless the associativity is violated.The Study-like determinant for the matrix representation of octonions still enjoys the composition law because quaternions obey the associativity.This proves the composition law for octonions.
A similar argument is applied to sedenions A 16 = S, treated in Sect. 5.It is shown that the Studylike determinant for the matrix representation of sedenions does not enjoy the composition law because octonions are no longer associative.The same argument is applied to general Cayley-Dickson algebras A 2 n with n > 4, which are discussed in Sect.6.The last section gives a short summary of the present work.
Appendix A discusses that the anti-commutativity among the linearly independent imaginary units defining hypercomplex numbers in dimensions ≥ 3 originates from the assumption that the imaginary part of the hypercomplex number is squared to be non-positive [4].We shall prove (33d) from (33a,b,c) for completeness in Appendix B. Schafer's proof [5] of the product rule for octonions, which makes use of the alternativity, will be reviewed in Appendix C. Some comments on Culbert's binary operation [6] are given in Appendix D.

Absence of Hamilton's triplet
Hamilton's triplet is defined by three real numbers x, y, z as where i, j are linearly independent imaginary units (i2 = j 2 = −1) commutable with real ones.The equality and the sum of triplets are defined as follows. x On the other hand, the product is obtained by the distributive law: Since the unit element 1 is independent of {i, j} and the real part of the triplet is squared to be non-negative (x 2 ≥ 0), the imaginary part of the triplet is squared to be non-positive: The reason for this is as follows. 2 If (iy + j z) 2 > 0, then there must exist a real number w = 0 such that (iy + j z) 2 = w 2 .Hence, we have w 2 − (iy + j z) 2 = [w − (iy + j z)][w + (iy + j z)] = 0, since the imaginary units are assumed to commute with real ones.According to the divisibility, we then have w = 1 • w = ±(iy + j z), which contradicts the assumption that {1, i, j} are linearly independent.Now the equality in (3) holds true if and only if we have y = z = 0.This is because the equality in (3) is valid if and only if we have iy + j z = 0, since the divisibility implies a = 0 if a 2 = 0. Since {i, j} are independent, y = z = 0 follows.Thus, we have Therefore, if we put Eq. ( 4) leads to i 2 ± (i j + ji) + j 2 = −2 ± 2α < 0 so that α is real and |α| < 1. Since, if α = 0, {i, j} already anti-commute with each other, we assume α = 0 in what follows.Put In addition to I 2 = −1 we require that J 2 = −1 and I J + J I = 0 hold true.That is, double signs correspond).This linear transformation was given by Dickson [7].Remember that it is not an orthogonal transformation but is given by triangular matrices.Because α ∈ R satisfies |α| < 1, a and b are real.Consequently, if we use {I, J } instead of {i, j}, the imaginary units of Hamilton's triplet become anti-commutative with each other.That is, the imaginary units {i, j} of (1) can be assumed to anti-commute from the outset without loss of generality: Then (i j) 2 = −1 from the associativity so that, for real y, z, we have Although there exists a solution to (8b), multiplying (8a) by i from the left and using the associativity lead to the last of which is not satisfied for any real z.Consequently, Hamilton's triplet closed under multiplication cannot exist if associativity and divisibility are assumed.The last stage of the above proof using (8a) and ( 9) is only a simplification of May's classroom note [3].
It is interesting, however, to note that the fact that i j is independent of {i, j} implies that i j does not commute but anti-commutes with {i, j}.Consequently, if there exists an additional imaginary unit k that anti-commutes with {i, j} but commutes with the product i j, then we would have consistent hypercomplex numbers that form a division algebra-this is nothing but Hamilton's quaternions.We shall see this more precisely in the next section.

Hamilton's quaternions from the anti-commutativity of three independent imaginary units
Let us now consider a 4D hypercomplex number where x, y, z, w are real and i, j, k are linearly independent imaginary units (i commuting with real numbers.The equality, sum, and product of these hypercomplex numbers are defined as for triplets in the previous section.The same is also true for other hypercomplex numbers, which will appear in the following sections, although we shall consider non-associative products for more than 2 factors there.It is now obvious that is valid, 3 where the equality holds true if and only if y = z = w = 0 are satisfied, thanks to the divisibility assumed here too.Consequently, we have so that, as in the previous section, the following equations are valid: where α, β, γ are real.We will see in the following that these α, β, γ can be set to zero without loss of generality.
Our derivation of quaternions never refers to the composition law, but it is, of course, true that quaternions furnish a composition algebra.
(26) Now, it is well known that any quaternion is Pauli-represented by a 2 × 2 matrix as where we set i = e 1 , j = e 2 , k = e 3 .Since the matrix (27a) is a complex one with e 2 3 = −1, its determinant immediately gives the squared norm of a quaternion q: Since ρ P (q)ρ P ( p) = ρ P (qp), we have the compostion law It turns out, however, that the representation (27a) using a complex matrix cannot be generalized to octonions because the latter can never be represented by (associative) quaternion-valued matrices (next generalization of (27a)) due to the non-associativity of octonions. 4 The representation is more suitable for that purpose, since the matrix ρ(q) is not a complex but a quaternion-valued one. 5It is equivalent to the Pauli representation: so that q → ρ(q) is really a representation with ρ(q)ρ(q ) = ρ(qq ).To compute the determinant of a quaternion-valued matrix ρ(q), we employ the Study determinant [2] Sdet 4 If one generalizes the Pauli representation (27a) to octonions, the composition of the two matrices requires a complicated operation due to non-associativity, since a simple matrix multiplication fails to reproduce the correct answer.See Appendix D. 5 It seems to be redundant to represent a quaternion in terms of a quaternion-valued matrix.The point is, however, that the quaternion-valued matrix representing a quaternion belongs to a special class of quaternionvalued matrices, which possess the usual well defined determinant, and the representation is easily generalized to octonions and so on.This generalization, in fact, starts from the following well known examples.A real number x is represented by a 2 × 2 matrix, ρ(x) = x 0 0 x .Similarly, a complex number z = x + iy for real x, y is represented by a 2 × 2 matrix, ρ P (z) = x −y y x or ρ(z) = x iy iy x .In the latter case, the determinant should be defined as (32).Apparently, both lead to the composition property of real and complex numbers.

7/19
Since the Study determinant satisfies the composition law, we again prove the composition law of quaternions.It can be noted, however, that, due to a special form of the matrix ρ(q), the same result can also be obtained as follows in terms of a usual determinant: where we have used the fact that the matrix ρ(q) is symmetric so that ρ † (q) ≡ ρ(q) T = ρ(q) = ρ( q).We refer this to as the Study-like determinant in the following.

Octonions
Let us now suppose that there exists an extra imaginary unit l(l 2 = −1) independent of Hamilton's ones {i, j, k}.Then, it follows from Here, a, b, c are real (except for |a|, |b|, |c| < 1, this is also proved by Dickson [7]).However, as in Sect.3, it can be shown that they can be set to zero without loss of generality (see Appendix B).Consequently, such l, if it exists, should satisfy the relation However, since k = i j, we have, due to the associativity, lk = l(i j) = (li) j = −(il) j = −i(l j) = i( jl) = (i j)l = kl (33e) so that lk = 0.If we assume the divisibility, l = 0 because k = 0 (we are going to define octonions as a division algebra).This is the content of the Frobenius theorem in algebra, which states that only divisible linear associative algebras over the real field are limited to R, C, and H. Consequently, we have to abandon the associativity to allow such l to anti-commute with {i, j, k}.It is easy to see that the above conclusion is avoided by replacing the associative calculation (33e) with the following: where the second, fourth, and sixth equalities violate the associativity.It is necessary, however, to retain the associativity in the following calculation: ensuring that i 2 = −1 commutes with the new imaginary unit l.Since we are looking for an algebra closed under multiplication, we have to include {li, l j, lk} as independent imaginary units.Consequently, the dimension of the hypercomplex number that we are going to define is 1 , the same as that of octonions.To derive octonions along this line it is necessary to require that all independent imaginary units anti-commute with each other.The anti-commutativity is a consequence of the assumption that the linear combination of the imaginary units {i, j, k, li, l j, lk, l} with real coefficients is squared to be non-positive (see Appendix A).Thus we have to set up a new calculational rule to ensure the anti-commutativity for the set {i, j, k, li, l j, lk, l}.The rule must violate the associativity as seen above but the violation must be only partial as (34b) obeys the associativity.Remembering that (1, i, j, k) = (1, e 1 , e 2 , e 3 ) and putting we should retain the associativity to ensure the imaginary nature of E i (i = 1, 2, 3): Since the difference between (35c) and (34a,b) lies in the number of l involved, we establish the following calculational rule (i, j, k in the following up to (40d) take the values 1, 2, 3): when only one of the new imaginary unit l is involved in the product e i (e j l) = (e j e i )l (36a) when 2 l are involved in the product l(le j ) = l 2 e j = −e j , (e j l)l = e j l 2 = −e j (le i )(e j l) = l(e i e j )l = l(−δ i j + i jk e k )l = +δ i j + i jk e k = −e j e i .(36b) We call this rule the "quasi-associative law" because it is a mixture6 of the associativity as in the i = j case of (36a) and in (36b), and the violation of the associativity as in the i = j case of (36a) ( i jk in (36b) is Levi-Civita's anti-symmetric tensor with 123 = 1).Here and hereafter, we employ Einstein's summation convention.
As noted earlier, the seven imaginary units {e 1 , e 2 , e 3 , E 1 , E 2 , E 3 , l} should anti-commute with each other: Equations (37a) are ensured by Hamilton's relations (24), which read and the associativity (36b) leading to Moreover, according to the quasi-associativity (36a) we have7 (l = e 7 ) e i E j = e i (le j ) = −e i (e j l) = −(e j e i )l = −(−δ ji + jik e k )l = δ i j e 7 − i jk E k (38c) E j e i = (le j )e i = l(e i e j ) = l(−δ i j + i jk e k ) = −δ i j e 7 + i jk E k .(38d) We also have from (36b).Consequently, we have verified (37a) and (37b) from the quasi-associative law (36a) and (36b), assuming the existence of l.It is true, moreover, that the quasi-associative law (36a) and (36b) not only assures the anti-commutativity (37) but also fixes the algebraic structure of octonions.Namely, {1, e i , E i = e 7 e i , e 7 } i=1,2,3 is the basis of octonions with (38) giving the multiplication table of octonions.To see this more compactly, we put Y = p + e 7 P, p = p 0 + p i e i , P = P 0 + P i e i ∈ H (39b) so that XY = (q + e 7 Q)( p + e 7 P) = qp + (e 7 Q) p + q(e 7 P) + (e 7 Q)(e 7 P) since, using the quasi-associative law (36a,b), we have q(e 7 P) = (q 0 + q i e i )[(e 7 (P 0 + P j e j )] = e 7 (q 0 P 0 ) + e 7 (q 0 P j e j ) + e 7 (−q i e i P 0 ) − e 7 q i P j (e i e j ) = e 7 ( q P) (40b) = e 7 ( p 0 Q 0 ) + e 7 ( p j e j Q 0 ) + e 7 ( p 0 Q i e i ) + e 7 ( p j Q i e j e i ) = e 7 ( pQ) (40c) It is well known that the product rule (40a) can be obtained by either the composition law or the alternative property of octonions (see Appendix C for Schafer's proof [5] based on the alternativity).
We herewith emphasize that they can also be derived by the quasi-associative law, which ensures the anti-commutativity between {i, j, k, li, l j, lk, l}.This is a generalization of our previous proof of Hamilton's formula i jk = −1 from the anti-commutativity between {i, j, k} to non-associative algebra.This implies that the Cayley-Dickson product for any A 2 n is not an assumption as in the literature but can be derived from the quasi-associative law, which replaces the associative law for n = 2.
In analogy to the new representation (28) of quaternions, we map an arbitrary octonion to a 2 × 2 octonion-valued matrix as follows: This is homomorphic if we assume the usual matrix product with the product of the matrix elements being subject to the quasi-associative law.That is, ρ(X )ρ(Y ) = q e 7 Q e 7 Q q p e 7 P e 7 P p = qp + (e 7 Q)(e 7 P) q(e 7 P) + (e 7 Q) p (e 7 Q) p + q(e 7 P) (e 7 Q)(e 7 P) + qp = qp − P Q e 7 ( q P + pQ) e 7 ( q P + pQ) where use has been made of (40b,c,d), which are the consequence of the quasi-associativity.The Study-like determinant is given, in analogy to (32), by where 1 2 is the 2D unit matrix.To confirm the composition law of octonions, we evaluate

Sedenions
Put {e i , E i = e 3+i , e 7 } i=1,2,3 = {e A } A=1,2,..., 7 .Then an octonion (39a) is rewritten as As in the case of octonions, such m exists only if the associativity is violated in addition to the non-associativity of octonions.It turns out, however, that it is enough to generalize the quasiassociativity (36) in an inductive way to ensure the anti-commutativity for the set of imaginary units {e A , me A , m} as follows: when only one of the new imaginary unit m is involved in the product a αβγ is totally antisymmetric with respect to the indices (αβγ ) and takes the values ± 1, 0 only (49c) so that {1, e α } α=1,...,15 are the linearly independent bases of sedenions S: For sedenions we find by using the quasi-associative law (45a) and (45b) that their product is given by since, as in the previous section, Here, the conjugation of X = x 0 + e A x A turns out to be X = x 0 − e A x A = q − e 7 Q.
We now define the map The reason for which Re [r ] = Re [s ] is the non-associativity of octonions.Thus sedenions fail to satisfy the composition law.In a sense, this is obvious from the fact that there are zero divisors (see, e.g., Ref. [9]) so that U V = 0 for some U, V = 0 when n > 3.
As in the case of octonions, the matrix product is non-associative.This completes a detailed form of Schafer's proof [5] of the product rule (55) for octonions, in a different notation.It is obvious that it depends heavily on the alternativity of octonions so that it can never be applied to sedenions and more general Cayley-Dickson algebras that are no longer alternative but flexible algebras with (U, V, U ) = 0 for U, V ∈ A 2 n when n > 3.

Appendix D. Some comments on Culbert's binary operation
Since This is an associative quaternion-valued matrix, whereas O is non-associative so that it cannot reproduce the correct multiplication law of octonions: ρ P (X )ρ P (Y ) = q −Q Q q p −P P p = qp − Q P −q P − Q p Qp + q P −Q P + q p = ρ P (XY ).
and the product rule (38a-e) is put into the formula (A, B, C = 1, 2, . . ., 7 up to (51b)): e 0 e A = e A e 0 = e A e A e B = −δ AB e 0 + a ABC e C a ABC is totally antisymmetric with respect to the indices (ABC) and takes the values ± 1, 0 only.(43b) PTEP 2014, 013A03 K. Morita Let m be an imaginary unit anti-commuting with e A : e A m + me A = 0. (44)

(
me A )e B = m(e B e A ) e A (e B m) = (e B e A )m (45a) when 2 m are involved in the product ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ m(me A ) = m 2 e A = −e A , (e A m)m = e A m 2 = −e A (me A )(e B m) = m(e A e B )m = m[(−δ AB e 0 + a ABC e C )]m = +δ AB e 0 + a ABC e C = −e B e A .(45b)Then {e A , me A , m} anti-commute with each other:e A e B + e B e A = 0 (A = B) (46a) m(me A ) + (me A )m = 0 (46b) e A (me B ) + (me B )e A = 0(46c)(me A )(me B ) + (me B )(me A ) = 0 (A = B) (46d)from (43b), the first equations of (45b), and e A (me B ) = −e A (e B m) = −(e B e A )m = δ AB m + a B AC (me C ) (47a) (me B )e A = m(e A e B ) = −δ AB m + a ABC (me C ) (47b) (me A )(me B ) = −(me A )(e B m) = −m(e A e B )m = −m(−δ AB + a ABC e C )m = −δ AB − a ABC e C (47c) because a ABC = −a B AC .Putting me A = E A and m = e 15 , we have e A e B = −δ AB + a ABC e C (48a) e A E B = δ AB e 15 − a ABC E C (48b) E B e A = −δ B A e 15 + a ABC E C (48c) E A E B = −δ AB e 15 − a ABC e C (48d) e A e 15 = −e 15 e A (48e) E A e 15 = e A .(48f ) This is obtained from (38a-e) by replacing i, j, k, l = e 7 , i jk → A, B, C, m = e 15 , a ABC .Now set {e A , E A , e 15 } A=1,...,7 = {e α } α=1,...,15 .Then, (48a-f) become (1 = e 0 being the unit element) e 0 e α = e α e 0 = e α (α = 1, 2, . . ., 15)