Is octonion algebra a subset of Clifford Algebra and if so which one? The relationship between division and Clifford algebras is explaned on this page.
We know that:
- Complex numbers are the even subalgebra of the Geometric Algebra based on 2D vectors (G+2,0) - see box on right for explanation of this terminology.
- Quaternions are the even subalgebra of the Geometric Algebra based on 3D vectors (G+3,0).
So is it possible that we could continue this sequence and say that octonions are the even subalgebra of the Geometric Algebra based on 4D vectors
(G+4,0)?
I think not because I have compared the multiplication tables of these algebras and they differ in at least two respects:
- octonions are not associative but G+4,0 is.
- I can't find a Geometric Algebra where all terms apart from the scalar term square to -ve
At this stage these conclusions are tentative because I have not validated the program which generated these results (see following section).
So it is looking like the above sequence ends with quaternions. If so how do octonions relate to multivectors? The only clue I can find comes from this paraphrase from the John Baez paper:
"Relationship between vectors and spinors holds true in 3,4,6 and 10 dimensions (one more than the dimensions of R,C,Q and O which gives the following isomorphisms:
SL(2,R) ≡SO(2,1)
SL(2,C) ≡SO(2,3)
SL(2,Q) ≡SO(2,5)
SL(2,O) ≡SO(2,9)"
where:
- SL(n, F) = The special linear group, consisting of n×n matrices where each element is of type 'F' with determinant =1.
- SO(n,R) = subgroup of E+(n), which consists of direct isometries, i.e., isometries preserving orientation; it contains those which leave the origin fixed. It is the rotation group of the sphere and all objects with spherical symmetry, if the origin is chosen at the center. Every orthogonal matrix has determinant either 1 or −1. The orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O(n,F) known as the special orthogonal group SO(n,F).
If SO(p,q) is defined with two numbers then I think it is the orthogonal group for any symmetric quadratic form Q with matrix signature (p,q). The group of matrices A which preserve Q, is denoted O(p,q). The Lorentz group is O(3,1).
Comparison of algebras using 'Evaluate' program
I have written a program which compares the multiplication tables (the essence of an algebra - for more information about these "Cayley" tables see this page) of different algebras, the program is described on this page.
The program generates the multiplication (Caley) table for a given algebra and can then do things like checking for commutivity and associativity, it does this by a sort of brute force method of trying all the possible combinations, if you know a more elegant mathematical method please let me know.
I have not properly validated this program so all these results should be treated as tentative. It is very difficult, take the associativity test, the progam tests (a* b)*c and a*(b * c) where we test every combination of a,b and c each of which, in this case, can have a value between 0 and 7 giving 8*8*8=512 combinations. If any of these are not associative then this is reported and its easy to check manually, but if the program goes through all combinations and finds the table associative, but how do I confirm that one of these combinations is not wrong?
Also the program only checks the commutivity and associativity of the basis values on their own, if does not check the commutivity and associativity of linear sums of these which make up a complete multivector.
I would like to extend this program to directly compare two multiplication tables. This is quite difficult because we not only need to compare the tables as they stand but also all combinations of swapping and inverting terms.
Octonion
This is the table often given for octonions:| e | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
| e1 | -e | e4 | e7 | -e2 | e6 | -e5 | -e3 |
| e2 | -e4 | -e | e5 | e1 | -e3 | e7 | -e6 |
| e3 | -e7 | -e5 | -e | e6 | e2 | -e4 | e1 |
| e4 | e2 | -e1 | -e6 | -e | e7 | e3 | -e5 |
| e5 | -e6 | e3 | -e2 | -e7 | -e | e1 | e4 |
| e6 | e5 | -e7 | e4 | -e3 | -e1 | -e | e2 |
| e7 | e6 | e6 | -e1 | e5 | -e3 | -e2 | -e |
analysing commutivity:
table does not commute:
for example: e1*e2 != e2*e1
analysing associativity:
table does not associate,
for example,
(e1* e2)* e3=e4* e3=-e6
is not equal to
e1*(e2* e3)=e1*e5=e6
how these results were generated.
Octonion (rearranged)
The table can be rearranged to give a table which looks closer to the Clifford algebras for easier comparison:| e | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
| e1 | -e | -e3 | e2 | -e5 | e4 | e7 | -e6 |
| e2 | e3 | -e | -e1 | -e6 | -e7 | e4 | e5 |
| e3 | -e2 | e1 | -e | -e7 | e6 | -e5 | e4 |
| e4 | e5 | e6 | e7 | -e | -e1 | -e2 | -e3 |
| e5 | -e4 | e7 | -e6 | e1 | -e | e3 | -e2 |
| e6 | -e7 | -e4 | e5 | e2 | -e3 | -e | e1 |
| e7 | e6 | -e5 | -e4 | e3 | e2 | -e1 | -e |
analysing commutivity: table does not commute:
for example: e1*e2 != e2*e1
analysing associativity:
table does not associate,
for example,
(e1* e2)* e4=-e3* e4=e7
is not equal to
e1*(e2* e4)=e1*-e6=-e7
how these results were generated.
Multivector 4D Even
This is based on a 4D vector space where all these dimensions square to +ve (4D euclidean space). We then take the even sub-algebra which contains only scalar, bivector and psudoscalar terms. The bivectors square to -ve because we have to swap two terms to simplify, for example:
(e1 ^ e2) ^ (e1 ^ e2) = -e1 ^ e1 ^ e2 ^ e2 = -e ^ e = -e
However the e1234 term squares to +ve, using the same rules, so this algebra can't be isomorphic to octonions which has 7 dimensions which square to -ve.
| e | e12 | e31 | e23 | e41 | e42 | e43 | e1234 |
| e12 | -e | e23 | -e31 | -e42 | e41 | -e1234 | e43 |
| e31 | -e23 | -e | e12 | e43 | -e1234 | -e41 | e42 |
| e23 | e31 | -e12 | -e | -e1234 | -e43 | e42 | e41 |
| e41 | e42 | -e43 | -e1234 | -e | -e12 | e31 | e23 |
| e42 | -e41 | -e1234 | e43 | e12 | -e | -e23 | e31 |
| e43 | -e1234 | e41 | -e42 | -e31 | e23 | -e | e12 |
| e1234 | e43 | e42 | e41 | e23 | e31 | e12 | e |
analysing commutivity:
table does not commute: for example: e1*e2 != e2*e1
analysing associativity: table associates
how these results were generated.
So this is another reason that this algebra is not isomorphic to octonions because octonions are not associative.
Multivector 4D Even(1 dimension squares to -ve)
This is similar to the situation above except one of the dimensions (e1) squares to -ve. Using the same rules we can see that the e1234 term now squares to -ve as required, however all bivector bases containing the e1 term now square to +ve, for example:
(e1 ^ e2) ^ (e1 ^ e2) = -e1 ^ e1 ^ e2 ^ e2 = -(-e) ^ e = +e
So, again, this algebra can't be isomorphic to octonions which has 7 dimensions which square to -ve.
| e | e12 | e31 | e23 | e41 | e42 | e43 | e1234 |
| e12 | e | -e23 | -e31 | e42 | e41 | -e1234 | -e43 |
| e31 | e23 | e | e12 | -e43 | -e1234 | -e41 | -e42 |
| e23 | e31 | -e12 | -e | -e1234 | -e43 | e42 | e41 |
| e41 | -e42 | e43 | -e1234 | e | -e12 | e31 | -e23 |
| e42 | -e41 | -e1234 | e43 | e12 | -e | -e23 | e31 |
| e43 | -e1234 | e41 | -e42 | -e31 | e23 | -e | e12 |
| e1234 | -e43 | -e42 | e41 | -e23 | e31 | e12 | -e |
Multivector 4D Even(2 dimensions squares to -ve)
In this case e1234 squares to +ve, also any bivector terms containing e1 or e2 (but not both) square to +ve. So no match here.
| e | e12 | e31 | e23 | e41 | e42 | e43 | e1234 |
| e12 | -e | -e23 | e31 | e42 | -e41 | -e1234 | e43 |
| e31 | e23 | e | e12 | -e43 | -e1234 | -e41 | -e42 |
| e23 | -e31 | -e12 | e | -e1234 | e43 | e42 | -e41 |
| e41 | -e42 | e43 | -e1234 | e | -e12 | e31 | -e23 |
| e42 | e41 | -e1234 | -e43 | e12 | e | -e23 | -e31 |
| e43 | -e1234 | e41 | -e42 | -e31 | e23 | -e | e12 |
| e1234 | e43 | -e42 | -e41 | -e23 | -e31 | e12 | e |
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1 analysing associativity: table associates
how these results were generated.
Multivector 4D Even(3 dimensions squares to -ve)
Bivector terms which contain e4 (the only term which squares to +ve) will square to +ve.
| e | e12 | e31 | e23 | e41 | e42 | e43 | e1234 |
| e12 | -e | -e23 | e31 | e42 | -e41 | -e1234 | e43 |
| e31 | e23 | -e | -e12 | -e43 | -e1234 | e41 | e42 |
| e23 | -e31 | e12 | -e | -e1234 | e43 | -e42 | e41 |
| e41 | -e42 | e43 | -e1234 | e | -e12 | e31 | -e23 |
| e42 | e41 | -e1234 | -e43 | e12 | e | -e23 | -e31 |
| e43 | -e1234 | -e41 | e42 | -e31 | e23 | e | -e12 |
| e1234 | e43 | e42 | e41 | -e23 | -e31 | -e12 | -e |
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1 analysing associativity: table associates
how these results were generated.
Multivector 4D Even(4 dimension squares to -ve)
The situation where all the vector dimensions square to -ve produces a table like the situation where all the vector dimensions square to +ve, in that, all the bivectors square to -ve but the e1234 term squares to +ve. Again no match.
| e | e12 | e31 | e23 | e41 | e42 | e43 | e1234 |
| e12 | -e | -e23 | e31 | e42 | -e41 | -e1234 | e43 |
| e31 | e23 | -e | -e12 | -e43 | -e1234 | e41 | e42 |
| e23 | -e31 | e12 | -e | -e1234 | e43 | -e42 | e41 |
| e41 | -e42 | e43 | -e1234 | -e | e12 | -e31 | e23 |
| e42 | e41 | -e1234 | -e43 | -e12 | -e | e23 | e31 |
| e43 | -e1234 | -e41 | e42 | e31 | -e23 | -e | e12 |
| e1234 | e43 | e42 | e41 | e23 | e31 | e12 | e |
analysing commutivity: table does not commute: for example: e1*e2 != e2*e1 analysing associativity: table associates.
