Grassmann Algebra

ExpTheGeneralizedProduct.nb 17
1����� 0 � k
�Β k
When Λ greater than zero, the A form is clearly zero, and hence the B form is zero also. Thus
we have the immediate result that
Β k
� Β 1
Λ
� k
Λ �
�
j�1
Β j
���� Β
k�Λ
j
Λ
� Β 1
k�Λ
We might express this more mnemonically as
� Β 1
Λ
Β
k
� Β 1
k�Λ
� Β 1
k�Λ
����  1
Λ
� Β 2
Λ
� Β 2
Λ
� Β 2
���� Β
k�Λ
2
Λ
� 0
� Β2 � �
k�Λ
� Β2 � Β
k�Λ
3 � Β
Λ
3 � �
k�Λ
� Β 2
���� Β
k�Λ
3
Λ
� � � 0
Generalized Grassmann products with the unit n-element
10.11
10.12
In a development mutatis mutandis of formulae 10.11 and 10.12 we determine the consequences
of one of the elements being the unit n-element 1
����� .
Suppose in the Generalized Product Theorem that Α is the unit n-element 1
m ����� .
�����
1 �����Λ �Β
k
� k
Λ �
� � � 1
����� ����  j
j�1
Λ
�� j
k�Λ
� k
Λ �
� � � 1
����� �  j
� ����  j
j�1
The interior product of the unit n-element with an arbitrary element is just the complement of
that element, since:
�����
1 ���� Βk � 1
����� ����� ����������� �����
� Β � 1 � Βk � Βk
k
Thus we see that when Λ is equal to k we have the trivial identity that:
�����
1 �����k �Β
k
�����
� Β
k
When Λ is less than k, the B form is clearly zero, and hence the A form is zero also. Note also
that the following expressions are equal, apart from a possible sign.
� 1
����� ����  j
Λ
�� j
k�Λ
� Β j
�����
j
� Β
Λ k�Λ
�������������
� Β j
Λ
����  j
k�Λ
Since if the complement of an expression is zero, then the expression itself is zero, we have
2001 4 26
k�Λ
Λ