Grassmann Algebra

TheInteriorProduct.nb 30
Interior product formulae for 1-elements from regressive product
formulae
Interior Product Formula 1
We start with the regressive product formula [3.41].
�Α � Β�� x � �Α � x ��Β � ��1�
m k n�1 m n�1 k
m �Α ��Β� x �
m k n�1
In this formula, put x equal to x
n�1 ����� (making x a 1-element) to get:
�Α � Β�� x
m k
����� � �Α � x
m ����� �� � ��1�
k
m �Α ��Β� x
m k
����� �
and then transform to the interior product form.
�Α � Β� ���� x � �Α ���� x��Β � ��1�
m k
m k
m�Α ��Β���� x�
m k
We have here rederived the fundamental product formula [6.64] of the last section.
Interior Product Formula 2
Now take formula 3.42 (the dual of 3.41).
�Α � Β��x � �Α � x��Β � ��1�
m k
m k
n�m Α ��Β� x�
m k
�����
Here, x is a 1-element. Put Β equal to Β and note that:
k
k
Α m ��Β k
� x� � ��1�n�1 �Α ��Β� x
m k
����� �����������
�
�Α ���� Β��x � �Α � x� ���� Β � ��1�
m k
m k
m�1 Α ���� �Β ���� x�
m k
We have here rederived the product formula [6.66] of the last section.
Interior Product Formula 3
An interior product of elements can always be expressed as the interior product of their
complements in reverse order.
Α ���� Β � ��1�
m k
�m�k���n�m� �Β
k
�����
����
�����
Αm
When applied to equation 6.69 above, the terms on the right-hand side interchange forms.
2001 4 5
6.69
6.70