Grassmann Algebra

TheExteriorProduct.nb 35
In 2-space therefore, all elements of � 0 , � 1 , and � 2 are simple. In 3-space all elements of � 0 , � 1 , � 2
and � 3 are simple. In higher dimensional spaces elements of grade 2 £ m £ nÐ2 are therefore the
only ones that may not be simple. In 4-space, the only elements that may not be simple are those
of grade 2.
There is a straightforward way of testing the simplicity of 2-elements not shared by elements of
higher grade. A 2-element is simple if and only if its exterior square is zero. (The exterior square
of Α m is Α m � Α m .) Since odd elements anti-commute, the exterior square of odd elements will be
zero, even if they are not simple. An even element of grade 4 or higher may be of the form of
the exterior product of a 1-element with a non-simple 3-element: whence its exterior square is
zero without its being simple.
We will return to a further discussion of simplicity from the point of view of factorization in
Chapter 3: The Regressive Product.
All (nÐ1)-elements are simple
In general, (nÐ1)-elements are simple. We can show this as follows.
Consider first two simple (nÐ1)-elements. Since they can differ by at most one 1-element factor
(otherwise they would together contain more than n independent factors), we can express them
as Α � Β1 and Α � Β2 . Summing these, and factoring the common (nÐ2)-element gives a
n�2 n�2
simple (nÐ1)-element.
Α
n�2 � Β1 � Α � Β2 � Α ��Β1 �Β2�
n�2 n�2
Any (nÐ1)-element can be expressed as the sum of simple (nÐ1)-elements. We can therefore
prove the general case by supposing pairs of simple elements to be combined to form another
simple element, until just one simple element remains.
� Conditions for simplicity of a 2-element in a 4-space
Consider a simple 2-element in a 4-dimensional space. First we declare a 4-dimensional basis
for the space, and then create a general 2-element Α. We can shortcut the entry of the 2-element
2
by using the GrassmannAlgebra function CreateBasisForm. The first argument is the grade
of the element required. The second is the symbol on which the scalar coefficients are to be
based. CreateBasisForm automatically declares the coefficients to be scalars by including
the new pattern a_ in the list of declared scalars.
�4; Α 2 � CreateBasisForm�2, a�
a1 e1 � e2 � a2 e1 � e3 � a3 e1 � e4 � a4 e2 � e3 � a5 e2 � e4 � a6 e3 � e4
Since Α 2 is simple, Α 2 � Α 2 � 0. To see how this constrains the coefficients ai of the terms of Α 2
we expand and simplify the product:
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