Grassmann Algebra

TheRegressiveProduct.nb 17
The duals of the above three cases are:
�
�
���Α m � Γ p
Α m � Β k
�Α � Β��Γ m k p
����
�
�
����Β
�
� ��� � 0 m�k�p�2�n�0 �k
p�
� �Α � Β��1 m k n
�Α� m ����Β
�k
�� n
�
� ��� ��
p�
n
� Application of the Common Factor Axiom
m � k � n � 0
m � k � p � 2�n � 0
3.24
3.25
3.26
Suppose we have two general 2-elements X and Y in a 3-space and we wish to find a formula for
their 1-element intersection Z. Because X and Y are in a 3-space, we are assured that they are
simple.
X � x1�e1 � e2 � x2�e1 � e3 � x3�e2 � e3;
Y � y1�e1 � e2 � y2�e1 � e3 � y3�e2 � e3;
We calculate Z as the regressive product of X and Y:
Z � X � Y
� �x1�e1 � e2 � x2�e1 � e3 � x3�e2 � e3�
� �y1�e1 � e2 � y2�e1 � e3 � y3�e2 � e3�
Expanding this product, and remembering that the regressive product of identical basis elements
is zero, we obtain:
Z � �x1�e1 � e2���y2�e1 � e3� � �x1�e1 � e2���y3�e2 � e3�
��x2�e1 � e3���y1�e1 � e2� � �x2�e1 � e3���y3�e2 � e3�
��x3�e2 � e3���y1�e1 � e2� � �x3�e2 � e3���y2�e1 � e3�
In a 3 space, regressive products of 2-elements are anti-commutative since
��1� �n�m� �n�k� � ��1� �3�2� �3�2� ��1. Hence we can collect pairs of terms with the same
factors:
Z � �x1�y2 � x2�y1���e1 � e2���e1 � e3�
��x1�y3 � x3�y1���e1 � e2���e2 � e3�
��x2�y3 � x3�y2���e1 � e3���e2 � e3�
We can now apply the Common Factor Axiom to each of these regressive products:
2001 4 5
Z � �x1�y2 � x2�y1���e1 � e2 � e3��e1
��x1�y3 � x3�y1���e1 � e2 � e3��e2
��x2�y3 � x3�y2���e1 � e2 � e3��e3