Abstract

An algebra closely related to the octonions, but with a distinguished triad, is described, and shown to be the only algebra (other than the octonions) composed of seven overlapping quaternionic subalgebras. The symmetry group preserving multiplication is shown to be a SU(2)xSU(2) subgroup of the G₂ symmetry group of the ordinary octonions, with vector-like transformation of the imaginary units of the distinguished triad, and and spinor-like transformation of the remaining four imaginaries. The natural choice of Dirac matrices can be extended to generate Lorentz transformations as well.

Much of the repetitive calculation was simplified by use of a few short C programs on a PC, which are presented and discussed. The techniques used in these programs are applicable to a wider range of problems, and could be the foundations of useful software tools for further octonionic research.

Introduction

A familiar representation of the octonions is given by the following multiplication table:
(1)

	e1	e2	e3	e4	e5	e6	e7 (2nd factor)
first factor
e1	-1	+e3	-e2	+e5	-e4	+e7	-e6
e2	-e3	-1	+e1	-e6	+e7	+e4	-e5
e3	+e2	-e1	-1	+e7	+e6	-e5	-e4
e4	-e5	+e6	-e7	-1	+e1	-e2	+e3
e5	+e4	-e7	-e6	-e1	-1	+e3	+e2
e6	-e7	-e4	+e5	+e2	-e3	-1	+e1
e7	+e6	+e5	+e4	-e3	-e2	-e1	-1

A line-by-line inspection reveals the well-known fact that each ej belongs to three quaternionic triads, with seven such triads in all. In the multiplication table given above, these triads are

(1a) (e₁,e₂,e₃), (e₁,e₄,e₅), (e₁,e₆,e₇), (e₂,e₅,e₇), (e₂,e₆,e₄), (e₃,e₄,e₇), and (e₃,e₅,e₆).

Tensor Notation

A compact notation for this same multiplication table is given by a totally antisymmetric 7x7x7 tensor e_ijk with the general properties:

(2) e_ijk = e_jki = e_kij = -e_ikj = -e_jik = -e_kji

(3) |e_ijk| = 1 iff (e_i,e_j,e_k) is a quaternionic triad, else e_ijk = 0.

(As will be seen, these conditions are necessary but not sufficient to define the octonions.) The multiplication table is given by the e_ijk's (and d_ij's such that d_ij = 1 if i = j, and d_ij = 0 if i not = j) thus:

(4) e_ie_j = -d_ij + e_ijke_k (summation over k).

A comment on notation: "*" will usually denote ordinary real multiplication, but may also be used for the octonionic product (as in the left-hand side of (4)) if necessary for legibility.

The specific form of the tensor for the example above is:

(5)
e₁₂₃ = e₂₃₁ = e₃₁₂ = -e₁₃₂ = -e₂₁₃ = -e₃₂₁ = 1
e₁₄₅ = e₄₅₁ = e₅₁₄ = -e₁₅₄ = -e₄₁₅ = -e₅₄₁ = 1
e₁₆₇ = e₆₇₁ = e₇₁₆ = -e₁₇₆ = -e₆₁₇ = -e₇₆₁ = 1
e₂₄₆ = e₄₆₂ = e₆₂₄ = -e₂₆₄ = -e₄₂₆ = -e₆₄₂ = -1
e₂₅₇ = e₅₇₂ = e₇₂₅ = -e₂₇₅ = -e₅₂₇ = -e₇₅₂ = 1
e₃₄₇ = e₄₇₃ = e₇₃₄ = -e₃₇₄ = -e₄₃₇ = -e₇₄₃ = 1
e₃₅₆ = e₅₆₃ = e₆₃₅ = -e₃₆₅ = -e₅₃₆ = -e₆₅₃ = 1

with all other e_ijk's = 0.

Choosing the Triads

There are in fact 30 ways of arranging 7 objects into 7 triads such that each pair of objects belongs to exactly 1 triad. Appendix A exhibits a computer program in c which generates the 30 possible multiplication tables like (1), corresponding to the 30 possible ways of choosing triads. Although all 30 representations are in principle equivalent, the representation given in (1) has the special property that if i, j, and k in the equation e_ie_j = (+/-)e_k are written in binary form, they satisfy k = i XOR j, where XOR is bitwise exclusive "or", and e₀ = 1. This is useful as a mnemonic.

The Norm and Nonassociativity

The -1 associated with the e₂₄₆ triad stems from the requirement that the octonion norm satisfy N(ab) = N(a)N(b); for octonion a given by

(6) a = a₀ + a_ie_i (summed over i),

the norm of a is given by

(7) N(a) = a₀a₀ + a_ia_i (summed over i).

It can also be related to the pattern of nonassociativity displayed by the octonionic imaginary units e_j, to wit:

(8) (e_ie_j)e_k = -e_i(e_je_k) iff (e_i, e_j, e_k) is not a quaternionic triad, and

(9) (e_ie_j)e_k = e_i(e_je_k) iff e_i, e_j, e_k are in the same triad.

This second condition is necessary if the triads are to be quaternionic.

At this point it is convenient to introduce a symbolic representation for multiplication which highlights the parallels to ordinary vector analysis. Represent the octonion a as a₀ + A; a₀ is the real part, and A the imaginary part, a vector with 7 components. The product of two octonions is given by

(10) ab = a₀b₀ + a₀B + b₀A +AxB - A.B,

where "x" represents a vector cross product defined by

(11) AxB = A_iB_je_ijke_k (all indices summed),

and "." represents the usual vector dot product A.B = A_iB_i (i summed). Straightforward calculation leads to the result that N(a)N(b) = N(ab) iff

(11a) (AxB).(AxB) = (A.A)(B.B) - (A.B)(A.B).

Adding a Twist

Something quite interesting happens if the -1 of the e₂₄₆ triad is changed to +1, resulting in the new multiplication table
(12)

	e1	e2	e3	e4	e5	e6	e7 (2nd factor)
first factor
e1	-1	+e3	-e2	+e5	-e4	+e7	-e6
e2	-e3	-1	+e1	+e6	+e7	-e4	-e5
e3	+e2	-e1	-1	+e7	+e6	-e5	-e4
e4	-e5	-e6	-e7	-1	+e1	+e2	+e3
e5	+e4	-e7	-e6	-e1	-1	+e3	+e2
e6	-e7	+e4	+e5	-e2	-e3	-1	+e1
e7	+e6	+e5	+e4	-e3	-e2	-e1	-1

with the associated e_ijk's

(13)
e₁₂₃ = e₂₃₁ = e₃₁₂ = -e₁₃₂ = -e₂₁₃ = -e₃₂₁ = 1
e₁₄₅ = e₄₅₁ = e₅₁₄ = -e₁₅₄ = -e₄₁₅ = -e₅₄₁ = 1
e₁₆₇ = e₆₇₁ = e₇₁₆ = -e₁₇₆ = -e₆₁₇ = -e₇₆₁ = 1
e₂₄₆ = e₄₆₂ = e₆₂₄ = -e₂₆₄ = -e₄₂₆ = -e₆₄₂ = 1
e₂₅₇ = e₅₇₂ = e₇₂₅ = -e₂₇₅ = -e₅₂₇ = -e₇₅₂ = 1
e₃₄₇ = e₄₇₃ = e₇₃₄ = -e₃₇₄ = -e₄₃₇ = -e₇₄₃ = 1
e₃₅₆ = e₅₆₃ = e₆₃₅ = -e₃₆₅ = -e₅₃₆ = -e₆₅₃ = 1.

This algebra no longer satisfies N(ab) = N(a)N(b), and has a pattern of associativity different from that of the octonions. It is still true that there are 7 quaternionic subalgebras, and that each e_j belongs to 3 of these 7 triads. However, exactly one of these triads is distinguished (in this example, it is (e₂,e₄,e₆)) by the fact that one obtains the octonions by reversing the signs of the e_ijk's for that triad. Reversing the signs of the e_ijk's of any one of the other triads will result, not in the octonions, but in an algebra with exactly one distinguished triad, the reversal of whose e_ijk's signs results in the octonions. In essence, there are but 2 algebras composed of 7 overlapping quaternionic subalgebras, namely, the octonions, and the "twisted" octonions having a distinguished triad.

Any algebra having 7 overlapping quaternionic triads (like the octonions and the "twisted" octonions) must satisfy the conditions (2) and (3). Thus, for a specific choice of triads, the only freedom in assigning the e_ijk's lies in choosing +1 or -1 for the right hand sides of each line of (13), giving 128 different choices in all. A convenient representation of the choice can be made using a 7-bit number: the lowest order bit is set if the first line of (13) is assigned -1, the next lowest bit is set if the second line of (13) is assigned -1, and so forth. It is straightforward to show that for the choice of triads given in (1a), condition (8) holds only for a sign assignment with 7-bit representation equal to one of the following:

(14) 8, 15, 17, 22, 34, 37, 59, 60, 67, 68, 90, 93, 105, 110, 112, or 119

or in hexadecimal notation (which is clearer for exhibiting the bit pattern):

(14a) 8, f, 11, 16, 22, 25, 3b, 3c, 43, 44, 5a, 5d, 69, 6e, 70, or 77.

Note that 16 choices of sign pattern give condition (8) for each of the 30 ways of grouping the e_j's into triads; thus the well-known fact that there are 480 representations of the octonions. Appendix B exhibits and discusses a c program used to find this result. In other words, the algebra is the octonions for these choices of signs of the e_ijk's. In particular, the multiplication table (1) corresponds to 7-bit sign value of 8. Moreover, every other integer between 0 and 127 can be obtained by changing one bit of exactly one of the numbers in (14). This bit clearly corresponds to the distinguished triad of the "twisted" octonions - reversing the signs of the e_ijk's associated with this triad gives a 7-bit sign value corresponding to the octonions, whereas reversing the signs for any other triad will give a 7-bit sign value not in (14), but one bit away from (exactly) one of the values in (14). Thus the assertion above that the octonions and the " twisted" octonions are the only 2 possible algebras satisfying (2) and (3) is seen to be true. Appendix B also gives a program "twist.c" which can be used to find the distinguished triad (if any) for an arbitrarily chosen 7-bit sign value.

There is a simple geometrical interpretation of the bit-changing properties described above. Any two distinct triads have exactly one element e_j in common, and each e_j belongs to three triads. If the sign bit is changed for two triads, there is a unique third triad containing the e_j common to the first two triads; changing the sign of this third triad as well is then equivalent to merely changing the sign of e_j, producing a (different) representation of the untwisted octonions.

To verify that the twisted octonions no longer satisfy N(ab) = N(a)N(b), expand (AxB).(AxB) explicitly, using the cross product definition (11). Cross terms involving the e_ijk's of 2 different triads will appear, and these must all cancel each other if the identity (11a) is to hold true. Inspection shows that the cross terms do indeed cancel only for the octonionic choices of sign for the e_ijk's. Thus, the twisted octonions do not satisfy N(ab) = N(a)N(b), and furthermore, the logical equivalence of the multiplicative norm and the octonions' associativity pattern is revealed.

An examination of the automorphism group of the twisted octonions reveals some interesting features. The general form of a linear transformation among the imaginary units is given by

(15) e'_j = C_jie_i (summed over i)

and the condition that such a transformation preserve multiplication is

(16) e'_ie'_j = e_ijke'_k - d_ij (summed over k)

which yields

(17) -C_imC_jm + C_ilC_jme_lmk = e_ijnC_nk - d_ij (summed over l, m, and n)

as necessary restrictions on the elements of C_ij. Since an automorphism must preserve the norm, C_ij must be orthogonal (i.e. C_imC_jm = d_ij); thus (17) gives

(18) C_ilC_jme_lmk = e_ijnC_nk (summed over l, m, and n).

A simple transformation satisfying conditions (18) corresponds geometrically to rotation of two triads, such that the imaginary element common to both triads is the axis of rotation, and the angles of rotation are equal and opposite, e.g.

(19)
e'₂ = e₂cos a + e₃sin a,
e'₃ = -e₂sin a + e₃cos a
e'₄ = e₄cos a - e₅sin a,
e'₅ = e₄sin a + e₅cos a

corresponds to a rotation around e₁, with e₂,e₃ rotated through an angle of +a, and e₄,e₅ rotated through -a. A second, independent rotation about e₁ mixes e₂ with e₃, and e₆ with e₇, with another arbitrary mixing angle. The third possibility, mixing e₄ with e₅ and e₆ with e₇, can be obtained by successive application of the rotations already described; hence, there are 2 independent parameters for rotations about e₁. For the usual octonions, any one of the imaginary units may be chosen as the axis of rotation for this type of elemental transformation, and the group generated by products of all such elemental transformations will thus have 14 independent parameters. This is the well-known automorphism group G₂.

It turns out, however, that the sign conditions imposed by the e_ijk's of the twisted octonions force the C_ij's to zero unless the axis of rotation lies in the distinguished triad. This generates an automorphism group corresponding to SU(2)xSU(2) rather than G₂.

There is an interesting and suggestive representation of the twisted automorphism group in which the distinguished triad looks very much like ordinary 3-dimensional space, and the remaining four imaginaries look like Dirac spinors with an extra internal SU(2) symmetry. To demonstrate this explicitly, the specific representation of the twisted octonions corresponding to 7-bit sign value 16 (10 hexadecimal) will be chosen. Changing the lowest order bit of 16 gives 17 (corresponding to one of the octonionic sign choices), showing that the distinguished triad is (e₁, e₂, e₃). We will thus consider rotations about e₁, e₂, and e₃. One such rotation with striking properties is that in which the angle of rotation within the distinguished triad e₁, e₂, e₃ is counterbalanced by opposite half-angle rotations about the same axis in each of the other 2 triads involving the unit chosen as rotation axis. More specifically, for rotation around e₁, the C_ij's are:

(20)

1	0	0	0	0	0	0
0	cos a	-sin a	0	0	0	0
0	sin a	cos a	0	0	0	0
0	0	0	cos a/2	sin a/2	0	0
0	0	0	-sin a/2	cos a/2	0	0
0	0	0	0	0	cos a/2	sin a/2
0	0	0	0	0	-sin a/2	cos a/2

For e₂ as rotation axis:
(21)

cos a	0	-sin a	0	0	0	0
0	1	0	0	0	0	0
sin a	0	cos a	0	0	0	0
0	0	0	cos a/2	0	sin a/2	0
0	0	0	0	cos a/2	0	-sin a/2
0	0	0	-sin a/2	0	cos a/2	0
0	0	0	0	sin a/2	0	cos a/2

(Note that e₂₅₇ = -1 in this representation). Finally, for e₃ as rotation axis:
(22)

cos a	-sin a	0	0	0	0	0
sin a	cos a	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	cos a/2	0	0	sin a/2
0	0	0	0	cos a/2	sin a/2	0
0	0	0	0	-sin a/2	cos a/2	0
0	0	0	-sin a/2	0	0	cos a/2