Grassmann Algebra

ExpTheGeneralizedProduct.nb 22
ToInteriorProducts�Α 4 ����� 2 �Α 4 � � 0
�Α1 � Α2 � Α3 � Α4 � Α1 � Α2��Α3 � Α4 �
�Α1 � Α2 � Α3 � Α4 � Α1 � Α3��Α2 � Α4 �
�Α1 � Α2 � Α3 � Α4 � Α1 � Α4��Α2 � Α3 �
�Α1 � Α2 � Α3 � Α4 � Α2 � Α3��Α1 � Α4 �
�Α1 � Α2 � Α3 � Α4 � Α2 � Α4��Α1 � Α3 �
�Α1 � Α2 � Α3 � Α4 � Α3 � Α4��Α1 � Α2 �� 0
ToInteriorProducts�Α 4 ����� 3 �Α 4 � � 0
�Α1 � Α2 � Α3 � Α4 � Α1 � Α2 � Α3��Α4 �
�Α1 � Α2 � Α3 � Α4 � Α1 � Α2 � Α4��Α3 �
�Α1 � Α2 � Α3 � Α4 � Α1 � Α3 � Α4��Α2 �
�Α1 � Α2 � Α3 � Α4 � Α2 � Α3 � Α4��Α1 �� 0
At the inner product level, some of the generalized products yield further relationships. For
example, Α 4 ����� 2 �Α 4 confirms the Zero Interior Sum Theorem.
Simplify�CollectTerms�ToInnerProducts�Α 4 ����� 2 �Α 4 ���
2 �Α1 � Α2 � Α3 � Α4 �Α1 � Α3 � Α2 � Α4 �Α1 � Α4 � Α2 � Α3� Α1 � Α2 � Α3 � Α4
Nilpotent products of non-simple elements
From the axioms for the exterior product it is straightforward to show that for a general (not
necessarily simple) m-element Α that Α � Α � ��1�
m m m m �Α � Α. This means, of course, that the
m m
exterior product of a general m-element with itself is zero if m is odd.
In a similar manner, the formula for the quasi-commutativity of the generalized product is:
Α m ����� Λ �Β k
� ��1� �m�Λ���k�Λ� �Β����� �Α
k Λ m
This shows that the generalized product of a general m-element with itself is:
Α m ����� Λ �Α m � ��1� �m�Λ� �Α m ����� Λ �Α m
Thus the generalization of the above result for the exterior product is that the generalized
product of a general m-element with itself is zero if m-Λ is odd.
2001 4 26
Α m ����� Λ �Α m � 0 m �Λ � OddIntegers
10.18