Maths - Cayley Table

This section discusses:

• Defining an algebra using a Cayley Table.
• Combining algebras by combining the individual Cayley tables.
• Application of Kronecker product and sum and .
• Spliting an algebra into subalgebras.
• Generating Clifford algebras and hyper-complex numbers by doubling up from simple 2 dimensional algebras.

Cayley tables allows us to easily lookup the result of multiplying any two elements of a given algebra. the result is made up of 3 parts:

• The scalar 'analogue' value of the result.
• Type of the result.
• Sign change of the result.

This assumes that the product of any two elements will be a single type which is the case for the simpler algebras that we are concerned with.

There are an infinite number of scalar values so we can't make a table from that but the type and sign can be combined into a table.

The Cayley Table is a good way to completely specify the multiplication rules of a multi-dimensional algebra. Since we are either working with algebras where we use only multiplication and not addition (groups), or we use algebras where addition is relatively trivial: the sum of corresponding terms, then effectively the whole algebra is defined by the Cayley table.

If we wan' t to work with more general type of addition then we would have to use the mathematical concept of a 'ring' but that's just a bit too general for me at the moment so I will assume that we are working with a 'vector space' which gives us relatively simple rules for addition and scalar multiplication:

• (v1,w1) + (v2,w2) = (v1+v2 ,w1+w2)
• α(v,w) = (αv,αw)

There may be some algebras where Cayley Tables are not applicable, that is: they don't have a table where multiplying each term produces one term, however Cayley Tables can be used for most of the algebras that we are interested in.

In the maths program that I am working on here each algebra is specified by its Cayley Table.

Relationship to Matrices

The Cayley Table itself is similar to a matrix, but operations such as addition and multiplication on the Cayley table aren't really useful so it is only really a pseudo matrix. However, the operations and are applicable.

What is the Cayley table for matrix algebra? Well we could define for matrices, but an n×n matrix would have n² elements, so the Cayley table would require n²×n² elements, so I think this is a case where using Cayley tables would not be very efficient. Doing matrix algebra in the conventional way is very easy to program for computers so this is only a problem where we want to mix matrix with other type of algebra.

Cayley Table

We can use a table to represent a finite group, but we can also use a Cayley Table to represent the multiplication rules of a group whose elements consist of a number of scalar dimensions, the table tells us how these dimensions combine.

For example, consider the group of complex numbers with the operation of multiplication,

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

The table only represents the type of each entry and whether there is any sign reversal, to actually multiply two we have to multiply the scalar values to determine there contribution to the given dimension.

The above table used '1' to represent the real numbers and 'i' to represent imaginary numbers but we could have used any symbols such as 'a' and 'b' it still fully defines the multiplication rules for complex numbers:

a*b	a	b
a	a	b
b	b	-a

The Cayley Table is similar to a Latin Square except that each symbol has sign (+ or -). I guess each symbol could be changed into two symbols although this would then double the dimensions of the table and this might hide the symmetries.

Cayley Table of a Group

The properties of a group result in the following properties when a cayley table represents a group

• no repetition in any row or column (how can I derive this?)
• one row and one column will represent the identity element and so will be in order.

Equivalence of Cayley Tables

How do we determine if two algebras are isomorphic be comparing their Cayley tables? It is not good enough to compare the two Cayley tables term by term because if we swap two terms or invert a given term then the algebras will still be isomorphic so we need to do a more complex comparison. This page explains how to test for equivilance of Cayley tables representing an isomorhism between the algebras.

Combining Cayley Tables

Can we combine Cayley Tables? For instance, imagine we have an algebra defined by table 1 and the elements of this algebra are another algebra defined by table 2. Can we combine these tables to give the overall table which defines the combined algebra?

On this page we look at various posibilities such as:

• Direct Sum and
• Direct Product

The Direct Product works for commutitive algebras (such as abelian groups) but most of the algebras we are working with have both commutitive and anti-commutitive terms so we can use the Direct Product to determine the type but use other means to determine the sign. So if we want to multiply:

• a e_i
• b e_j

For the type of algebras we are discussing, we could detemine the product of these terms by:

(a e_i)*(b e_j) = a*b*S_ij*(e_ie_j)

where:

• a, b = the scalar values.
• S_ij = represents the sign, it is a step function which is -1 or +1 depending on i and j.
• e_ie_j = represents the type given by the Kronecker product.

S_ij depends on the type of Algebra:

For Clifford algebras it is a Walsh function (or it can be worked out by the number of swaps to order the indicies as described on this page).

For hypercomplex numbers we can use the Cayley-Dickson process.

Matrix Algebra

Can we use these tables to represent matrix multiplication? This would not necessarily be an efficient way to implement the multiplication in a computer program for example. However it might be interesting as a basis for an algebra that is a superset of matrix algebra and other elements. It might also allow us to better define algebras which are a combination of matrix algebra and other elements such as complex numbers (Pauli Matrices).

If we want to multiply two matrices containing 4 elements:

e1	e2
e3	e4

Then we can represent the multiplication with the following table:

	b.e1	b.e2	b.e3	b.e4
a.e1	e1	0	e3	0
a.e2	e2	0	e4	0
a.e3	0	e1	0	e3
a.e4	0	e2	0	e4

So, although we think of a 2x2 matrix as a 2D matrix, it actually has 4 elements. Therefore, in these terms, it is a 4D quantity like quaternions.

Therefore, if we want to define the cayley table for an 'n x n' matrix then we need a table which has n² rows and columns. So for large matrices the cayley table would get very large.

Further Reading

Cayley Table for Geometric Algebra:

• Cayley Table for Geometric Algebra

Generating Geometric Algebras

Geometric Algebras Categories

Group Theory