Grassmann Algebra

ExploringHypercomplexAlgebra.nb 3
We could also explore the algebras generated by products of the type defined by
m,Λ,k
definition 11.1, but which have some of the Σ defined to be zero. For example the
relations,
m,Λ,k m,Λ,k
Σ � 1, Λ�0 Σ � 0, Λ�0
defines the algebra with the hypercomplex product reducing to the exterior product, and
m,Λ,k m,Λ,k
Σ � 1, Λ�Min�m, k� Σ � 0, Λ�Min�m, k�
defines the algebra with the hypercomplex product reducing to the interior product.
Both of these definitions however lead to products having zero divisors, that is, some
products can be zero even though neither of its factors is zero. Because one of the
principal characteristics of the hypercomplex product is that it has no zero divisors, we
shall limit ourselves in this chapter to exploring just those algebras. That is, we shall
m,Λ,k
always assume that Σ is not zero.
We begin by exploring the implications of definition 11.1 to the generation of various
hypercomplex algebras by considering spaces of increasing dimension, starting with a
space of zero dimensions generating the real numbers. In order to embed the algebras
defined on lower dimensional spaces within those defined on higher dimensional
m,Λ,k
spaces we will maintain the lower dimensional relations we determine for the Σ
when we explore the higher dimensional spaces.
Throughout the development we will see that the various hypercomplex algebras could
m,Λ,k
be developed in alternative fashions by various combinations of signs Σ and
metrics. However, we shall for simplicity generally restrict the discussion to
constraining the signs only and keeping products of the form ei ���� ei positive.
Although in this chapter we develop the hypercomplex product by determining more
m,Λ,k
and more constraints on the signs Σ that give us the properties we want of the
number systems �, �, �, and �, it can be seen that this is only one direction out of the
multitude possible. Many other consistent algebras could be developed by adopting
m,Λ,k
other constraints amongst the signs Σ .
ei ���� ei
11.2 Some Initial Definitions and Properties
The conjugate
The conjugate of a Grassmann number X is denoted X c and is defined as the body Xb
(scalar part) of X minus the soul Xs (non-scalar part) of X.
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