Maths - Cayley Table - 4D

On this page we will generate various 4 dimensional Cayley tables using combinations of the following methods:

The aim is to look for correspondences between the various types of algebras. We will see that, for these 4D algebras, that the Cayley-Dickson method produces the same algebras as the Clifford algebras:

When we double up to 8D algebras we will see that the algebras will start to diverge.

For two algebras to be isomorphic (that is there is a mapping between equivalent algebras) then the tables don't necessarily need to be identical, we can swap or invert any dimensions (see this page), so there are other equivalences that we can find. For instance:

D

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
e1 e e12 e2
e2 e12 e e1
e12 e2 e1 e
or equivalently in quaternion notation:
1 i j k
i 1 k j
j k 1 i
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:

4D Cayley-Dickson Algebras

 

D⊗D

D⊗C C⊗D C⊗C
i & j anticommute
i*j = -j*i
+ + + +
+ + + +
+ - + -
+ - + -
+ + + +
+ - + -
+ - + -
+ + + +
+ + + +
+ + + +
+ - - +
+ - - +
+ + + +
+ - + -
+ - - +
+ + - -
i & j commute
i*j = j*i
+ + + +
+ + + +
+ + + +
+ + + +
+ + + +
+ - + -
+ + + +
+ - + -
+ + + +
+ + + +
+ + - -
+ + - -
+ + + +
+ - + -
+ + - -
+ - - +

i & j anticommute

but left handed

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

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.

These results are labeled as follows:

Note that in this context⊗denotes the direct product as explained on this page (Its not the kronecker product of the tables).

4D Clifford Algebras

That is, Clifford algebras based on 2 vector dimensions, I have tried the combinations of these dimensions squaring to positive and negative.

 

both dimensions square to +ve

G 2,0,0

one dimension squares to -ve other to +ve

G 1,1,0

one dimension squares to +ve other to -ve

G 1,1,0

both dimensions square to -ve

G 0,2,0

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 8D Clifford Algebras

That is, An even subalgebra of Clifford algebras based on 3 vector dimensions.

 

G+ 3,0,0

G+ 2,1,0

G+ 1,2,0

G+ 0,3,0

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

how these results were generated.


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

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