Maths - Cayley Table - 8D

When we looked at 4D algebras we saw that the algebras derived from Cayley-Dickson method produces the same algebras as the Clifford algebras, but now when we double up to 8D algebras, we see that the two classes of algebra start to diverge.

We can see that the tables for the algebras as different and further checking shows that the algebras are not isomorphic. The Cayley-Dickson have a conjugate and are therefore dividable but they are not associative. The Clifford algebras are associative but they may not have an inverse.

As we have seen on this page the type of each entry will be common for all these methods, it will be:

e e1 e2 e12 e3 e13 e23 e123
e1 e e12 e2 e13 e3 e123 e23
e2 e12 e e1 e23 e123 e3 e13
e12 e2 e1 e e123 e23 e13 e3
e3 e13 e23 e123 e e1 e2 e12
e13 e3 e123 e23 e1 e e12 e2
e23 e123 e3 e13 e2 e12 e e1
e123 e23 e13 e3 e12 e2 e1 e
or equivalently in octonion notation:
1 i j k l li lj lk
i 1 k j li l lk lj
j k 1 i lj lk l li
k j i 1 lk lj li l
l li lj lk 1 i j k
li l lk lj i 1 k j
lj lk l li j k 1 i
lk lj li l k j i 1

So to make the comparison clearer on the page we will only show the sign and colour code the entries so that the pattern will show:

8D Cayley-Dickson Algebras

 

D⊗D⊗D

D⊗D⊗C D⊗C⊗C O = C⊗C⊗C
i*j = -j*i
+ + + + + + + +
+ + + + + + - -
+ - + - + + + +
+ - + - + + - -
+ - - - + - - -
+ - - - + - + +
+ + - + + - - -
+ + - + + - + +
+ + + + + + + +
+ - + - + - - +
+ - + - + + + +
+ + + + + - - +
+ - - - + - - -
+ + - + + + + -
+ + - + + - - -
+ - - - + + + -
+ + + + + + + +
+ - + - + - - +
+ - - + + + - -
+ + - - + - + -
+ - - - + - - -
+ + - + + + + -
+ + + - + - + +
+ - + + + + - +
+ + + + + + + +
+ - + - + - - +
+ - - + + + - -
+ + - - + - + -
+ - - - - + + +
+ + - + - - - +
+ + + - - + - -
+ - + + - - + -

how these results were generated.

As the above link explains, the table was generated by a computer program from the (modified) Caley-Dickson doubling process.

8D Clifford Algebras

That is, Clifford algebras based on 3 vector dimensions, I have tried the combinations of these dimensions squaring to positive and negative.

 

G 3,0,0

G 2,1,0

G 1,2,0

G 0,3,0

vectors anti-commute
+ + + + + + + +
+ + + + + + + +
+ - + - + - + -
+ - + - + - + -
+ - - + + - - +
+ - - + + - - +
+ + - - + + - -
+ + - - + + - -
+ + + + + + + +
+ - + - + - + -
+ - + - + - + -
+ + + + + + + +
+ - - + + - - +
+ + - - + + - -
+ + - - + + - -
+ - - + + - - +
+ + + + + + + +
+ - + - + - + -
+ - - + + - - +
+ + - - + + - -
+ - - + + - - +
+ + - - + + - -
+ + + + + + + +
+ - + - + - + -
+ + + + + + + +
+ - + - + - + -
+ - - + + - - +
+ + - - + + - -
+ - - + - + + -
+ + - - - - + +
+ + + + - - - -
+ - + - - + - +

how these results were generated.

As the above link explains, the table was generated by a computer program from the ordering of bases.

Even Subalgebras of 16D Clifford Algebras

That is, An even subalgebra of Clifford algebras based on 4 vector dimensions.

 

G+ 4,0,0

G+ 3,1,0

G+ 2,2,0

 
+ + + + + + + +
+ - - + - + + -
+ + - - - - + +
+ - + - + - + -
+ + + + - - - -
+ - - + + - - +
+ + - - + + - -
+ - + - - + - +
+ + + + + + + +
+ + + + + + + +
+ - + - + - + -
+ - + - + - + -
+ - - + + - - +
+ - - + + - - +
+ + - - + + - -
+ + - - + + - -
+ + + + + + + +
+ - + - + - + -
+ - + - + - + -
+ + + + + + + +
+ - - + + - - +
+ + - - + + - -
+ + - - + + - -
+ - - + + - - +
 

G+ 0,4,0

G+ 1,3,0

 

 
+ + + + + + + +
+ - + - + - + -
+ - - + + - - +
+ + - - + + - -
+ - - + - + + -
+ + - - - - + +
+ + + + - - - -
+ - + - - + - +
+ + + + + + + +
+ - + - + - + -
+ - - + + - - +
+ + - - + + - -
+ - - + + - - +
+ + - - + + - -
+ + + + + + + +
+ - + - + - + -
 

Further Reading

Cayley Table for Geometric Algebra:


metadata block
see also:

 

Correspondence about this page

Book Shop - Further reading.

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flag flag flag flag flag flag Deep Down Things: The Breathtaking Beauty of Particle Physics - If you dont want any equations then this is a good and readable introduction to quantum theory and related mathematics such as Lie groups, Gauge Theory, etc.

 

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Terminology and Notation

Specific to this page here:

 

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