Maths - Cayley Table - Subalgebras

A subset of a group is only a subgroup if, for every 'a' and 'b' then a*b and a-1 are also members of the group, for example, if we have the following group we could take a subset by removing the 'j' and 'k' rows and columns:

b.1 b.i b.j b.k
a.1 1 i j k
a.i i -1 k -j
a.j j -k -1 i
a.k k j -i -1

This gives:

b.1 b.i
a.1 1 i
a.i i -1

Which is a valid group because it only contains '1' and 'i' types. However, if we removing the '1' and 'k' rows and columns, then we don't get a valid group.

b.1 b.i b.j b.k
a.1 1 i j k
a.i i -1 k -j
a.j j -k -1 i
a.k k j -i -1

This contains 'k' which is not an element of the group:

b.i b.j
a.i -1 k
a.j -k -1

Even Subalgebras

This applies to Clifford algebras where the elements are grouped into blades (scalars,vectors,bivectors,trivectors…). To take an even subalgebra we halve the number of dimensions by keeping the even blades (scalars,bivectors…) and removing the odd blades (vectors,trivectors…).

As we have seen on this page there are advantages to numbering the elements in bit order even though this mixes up the blades. If we do this its still easy to determine what each element is by the number of '1's it has. When they are ordered in this way we can see that each pair of original dimensions produces one dimension in the even subalgebra:

original dimensions subalgebra dimensions new numbering
000 = scalar 000 = scalar 00
001 = vector
010 = vector 011 = bivector 01
011 = bivector
100 = vector 101 = bivector 10
101 = bivector
110 = bivector 110 = bivector 11
111 = trivector

So we can see that the Cayley table for the even subalgebra still has the checkerboard pattern that is common to the algebras that we are studying.

000 001 010 011 100 101 110 111
001 000 011 010 101 100 111 110
010 011 000 001 110 111 100 101
011 010 001 000 111 110 101 100
100 101 110 111 000 001 010 011
101 100 111 110 001 000 011 010
110 111 100 101 010 011 000 001
111 110 101 100 011 010 001 000

So the pattern of even subalgebras is the same as the Clifford algebras on which they are based, what changes is the sign of the terms.

Odd Subalgebras

Odd subalgebras are not closed so we don't tend to use them.

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