Maths - Multi-Dimensional Algebra

So far we have been looking at equations containing real numbers, in this section we will be looking at algebras where each element is a compound type made up of multiple real quantities. In geometric terms each of these real quantities represents a dimension. Most of the algebras are vector spaces where one of the dimensions will have exactly the same properties as real numbers, the other elements may interact with each other in a slightly different way, perhaps they may anti-commute instead of being commutative. We sometimes refer to these other elements as the non-real or imaginary dimensions and they often consist of a real number with an operator like 'i'.

If you are interested in a specific algebra such as vectors, complex numbers, quaternions and Clifford algebra then it is best to go directly to these topics now. Read this section if you want to investigate the common theory of these algebras.

One of the reasons for investigating this subject is to create a computer program which can work with different algebras and combine them in various ways, I am working on such a program, here.

To do this we will use the following approaches:

Defining an algebra using a Cayley Table.
Relating the Cayley table to algebraic properties such as commutative, associative, distributive, divisible, etc.
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.

Multi-Dimensional Algebra

Sometimes it makes sense to combine together a set of real numbers to form a single, indivisible entity, we can then use this entity in equations in the same way that we have used real number up to now. When we do this we need to be sure that we know the rules for working with these entities, the rules may possibly be different than when we are working with real numbers. Each algebra will also have its own rules for addition and/or multiplication and each will have its own notation. For more information see this page.

vectorSpace

For such a precise subject the terminoligy in mathematics is difficult to tie down. It seems that when mathemations extend the subject they tend to reinterpret what went before and we end up with overlapping terminoligy. I am trying to keep the terminoligy, on this site, consistant at much as I can: defintions on this page.

Vector Spaces

Vector spaces have at least one dimension that behaves like real algebra, known as a 'scalar' and it has simple rules for addition and scalar multiplication:

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

That is: the rule for addition is that we add corresponding terms, for scalar multipication: to multiplying the whole term by a scalar is equivalent to multiplying each element by the scalar.

In classical vector algebra the scalar is a seperate quantity from the vector quantity and there are 3 types of multipication defined:

scalar multipication: scalar×vector=vector
dot multipication: vector•vector=scalar
cross multipication: vector×vector=vector (only works in 3 & 7 dimensions).

K-Algebra

This is a vector space where there is a multipication wich can be applied to any terms and will alway give a valid result (multiplication is closed). This result is detemined by multiplying the basis elements.

Matricies

Matricies can represent any linear relationship. There is an isomorphism between certain matricies (orthogonal) and some hypercomplex algebras.

Hypercomplex Algebras

This is where the terminology is most vauge 'Hypercomplex numbers' tends to be used for various generalisations of complex numbers. Here we will use 'hypercomplex' to represent algebras detemine the product of two 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.

e_ie_j will always be the same for an algebra of a given dimension and can easily be calculated, provided the elements are represented in bit order, by taking the bitwise exclusive or of the two operands.

So the nature of the entire algebra can be represented by S_ijthe sign value of the product. For an 'n' dimensional algebra this can be shown by an n×n array of binary values. For a fuller explanation of this see this page.

The two most important examples of hypercomplex algebras are Clifford and Cayley-Dickson Algebras (which overlap to some extent) although there may be other hypercomplex algebras?

Clifford Algebras

In Clifford algebras the product of two different vector basis e_i^e_j cannot be simplified further and is left as it is as a bivector basis. In futher multipications we bay go on to generate trivector basis and so on.

Cayley-Dickson Algebras

In Cayley-Dickson algebras the product of two different vector basis e_i^e_j is represented as a new vector basis e_k and in any further iterations is treated as just another vector basis. Converting bivectors to vectors at each stage explains why the algebraic properties of these algebras degrade every time we double up the dimension.

More on the Cayley-Dickson doubling process here.

Cayley Table

The Cayley Table is a good way to completely specify the multiplication rules of a multi-dimensional algebra. This 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.

For more information see this page.

Terminology

We will be using the following notation on these pages:

notation	meaning	basis	Alternative
	direct sum

^	exterior algebra
^^k	exterior algebra k^th blade
	scalar	e0
²	2 dimensional vector	e1,e2
³	3 dimensional vector	e1,e2,e3
ⁿ	n dimensional vector	e1…en
^²²	bivector based on 2D algebra	e12
^²³	bivector based on 3D algebra	e12,e31,e23
C	complex numbers
H	quaternions		³
O	octonions		⁷