There is an alternative way to encode the Clifford algebras, that is to represent the bases e1,e2... and the higher order bases e12... as matrices. Any multivector can therefore be represented as a matrix. This matrix will be a linear combination of these bases.
| Note: this is different from a matrix transformation in like this: Unlike a conventional matrix transform a matrix representation behaves exactly equivalent to Clifford Algebra so transforms are of the form: m*p*m-1 |
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Representing groups using linear algebra like this is known as representation theory.
An example of this is Pauli Matricies which can be used to represent quaternion like algebra.
Applying Representation theory to Clifford Algebra
In this case we have:
v • u + u • v = 2 Q(v,u)
or in in terms of the basis vectors:
ei • ej + ej • ei = 2 Q(ei,ej)
where:
- v, u are vectors
- ej,ej are basis vectors
- Q is quadratic form
Generating a Clifford algebra using a program
Here I am experimenting with a computer algebra program to:
- Generate a group from the Clifford algebra
- Generate the representation from the group.
We can use a computer program to generate these groups, here I have used Axiom/FriCAS which is described here.
K := Fraction(Integer)
Type: Type
CA := CliffordAlgebra(2,K,[[1,0],[0,1]])
Type: Type
F := toFiniteGroup()$CA
toFiniteGroup # elements=8
generateNames n=8
toFiniteGroup strngs:11-11e1-e1e2-e2e12-e12
Type: Type
toTable()$F
Type: Table(8)
PG := toPermutation()$F
<(1 2)(3 4)(5 6)(7 8),(1 4)(2 3)(5 8)(6 7),(1 6)(2 5)(3 7)(4 8)>
Type: PermutationGroup(PositiveInteger)
permutationRepresentation(PG::(List Permutation PositiveInteger),8)
Type: List(Matrix(Integer))
(7) ->
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| if we replace |
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