This is a multipication table that defines the multipication operation for a given algebra. We can use a 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,
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.
This is different from the Cayley tables for finite groups (as explaned on the group theory pages) in that -1 appears in the body of the table but is not in the row or column identifiers. To make this into a valid Cayley table we need to have rows and columns for -1 and -i. We can easily do this by replacing the rows and columns as follows:
|1 is replaced by:||
|-1 is replaced by:||
|i is replaced by:||
We then get a valid Cayley table:
This now shows that it is a valid group. However, for most purposes, it is easier to use the smaller table with minus entries in the body of the table but not row and column headings represning the minus entries.
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 operationsandare 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.
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 Sumand
- Direct Product
The Direct Productworks 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 ei
- b ej
For the type of algebras we are discussing, we could detemine the product of these terms by:
(a ei)*(b ej) = a*b*Sij*(eiej)
- a, b = the scalar values.
- Sij = represents the sign, it is a step function which is -1 or +1 depending on i and j.
- eiej = represents the type given by the Kronecker product.
Sij 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.
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:
Then we can represent the multiplication with the following table:
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.
Cayley Table for Geometric Algebra: