Grassmann Algebra

ExploringGrassmannAlgebra.nb 17
This algorithm is used in the general power function GrassmannPower described below.
� Powers of Grassmann numbers with no body
A further simplification in the computation of powers of Grassmann numbers arises when they
have no body. Powers of numbers with no body will eventually become zero. In this section we
determine formulae for the highest non-zero power of such numbers.
The formula developed above for positive powers of Grassmann number still applies to the
specific case of numbers without a body. If we can determine an expression for the highest nonzero
power of an even number with no body, we can substitute it into the formula to obtain the
required result.
Let S be an even Grassmann number with no body. Express S as a sum of components Σ, where
i
i is the grade of the component. (Components may be a sum of several terms. Components with
no underscript are 1-elements by default.)
As a first example we take a general element in 3-space.
S �Σ�Σ 2 �Σ 3 ;
The (exterior) square of S may be obtained by expanding and simplifying S�S.
��S � S�
Σ � Σ 2 �Σ 2 � Σ
which simplifies to:
S � S � 2�Σ � Σ 2
It is clear from this that because this expression is of grade three, further multiplication by S
would give zero. It thus represents an expression for the highest non-zero power (the square) of
a general bodyless element in 3-space.
To generalize this result, we proceed as follows:
1) Determine an expression for the highest non-zero power of an even Grassmann number with
no body in an even-dimensional space.
2) Substitute this expression into the general expression for the positive power of a Grassmann
number in the last section.
3) Repeat these steps for the case of an odd-dimensional space.
The highest non-zero power pmax of an even Grassmann number (with no body) can be seen to
be that which enables the smallest term (of grade 2 and hence commutative) to be multiplied by
itself the largest number of times, without the result being zero. For a space of even dimension n
this is obviously pmax = n ���� 2 .
For example, let X be a 2-element in a 4-space.
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