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.
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 eijk with the general properties:
(2) eijk = ejki = ekij = -eikj = -ejik = -ekji
(3) |eijk| = 1 iff (ei,ej,ek) is a quaternionic triad, else eijk = 0.
(As will be seen, these conditions are necessary but not sufficient to define the octonions.) The multiplication table is given by the eijk's (and dij's such that dij = 1 if i = j, and dij = 0 if i not = j) thus:
(4) eiej = -dij + eijkek (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 eijk'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. A computer program may be requested from the authors 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 eiej = (+/-)ek 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 mneumonic.
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 + aiei (summed over i),
with a norm
(7) N(a) = a0a0 + aiai (summed over i).
It can also be related to the pattern of nonassociativity displayed by the octonionic imaginary units ej, to wit:
(8) (eiej)ek = -ei(ejek) iff (ei, ej, ek) is not a quaternionic triad, and
(9) (eiej)ek = ei(ejek) iff ei, ej, ek 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:
(10) ab = a0b0 + a0B + b0A +AxB - A.B,
where "x" represents a vector cross product defined by
(11) AxB = AiBjeijkek (all indices summed),
and "." represents the usual vector dot product A.B = AiBi (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 eijk'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 ej 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 eijk's for that triad. Reversing the signs of the eijk'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 eijk'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 eijk'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.
A c program may be obtained from the authors upon request which is used to find this result. In other words, the algebra is the octonions for these choices of signs of the eijk'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 eijk'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. A program "twist.c" may also be requested from the authors, which can be used to find the distinguished triad (if any) for an arbitrarily chosen 7-bit sign value.
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 eijk'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 eijk'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 = Cjiei (summed over i)
and the condition that such a transformation preserve multiplication is
(16) e'ie'j = eijke'k - dij (summed over k)
which yields
(17) -CimCjm + CilCjmelmk = eijnCnk - dij (summed over l, m, and n)
as necessary restrictions on the elements of Cij. Since an automorphism must preserve the norm, Cij must be orthogonal (i.e. CimCjm = dij); thus (17) gives
(18) CilCjmelmk = eijnCnk (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 eijk's of the twisted octonions force the Cij'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.
Thus the twisted octonion algebra leads to a natural choice of Dirac matrices which are then extended to generate Lorentz transformations as well [1]. The intimate ties of these octonionic structures to Hopf and pseudo-Hopf fibrations, connections to higher dimensional solotonic models, Kaluza-Klein compactification of D=11 supergravity, GUTs, superstrings, and supermembrane theories will be discussed in another publication. For a review we refer the reader to some recent articles shown in reference 2. References 3 and 4 contain some of their applications to hadronic color algebra.