Maths - Cayley Table - 4D

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

• From complex and double numbers combined using the (modified) Cayley-Dickson method.
• From Clifford algebras based on 2D vector algebra (each of which may square to +ve or -ve).
• From even subalgebras of 8D clifford algebras (again using various combinations of +ve and -ve squaring basis).

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:

• DD = G 2,0,0
• DC = G 1,1,0
• CD = G 1,1,0
• CC = G 0,2,0

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:

DC CD G 1,1,0

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

	DD	DC	CD	CC
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:

• DD = Double where each element is a Double
• DC = Double where each element is a Complex
• CD = Complex where each element is a Double
• CC = Complex where each element is a Complex

Note that in this contextdenotes 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.

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Group Theory