Grassmann Algebra

TheComplement.nb 41
e i
m
�����
i � ej � e
m m
In � 1 these reduce to:
ei
m
jk ����� i
��gm �ek�
� e
m m
�
����� m
ej � gij� 1
m
�����
ei � e j
����� j �����
�∆i� 1
ei � ej
����� � gij� 1
�����
� gm jk
�� �����
The regressive product of basis elements
g ��ek ij
� gm � 1
����� m
�����
e i
m
����� m
� g ij
� 1
�����
� ej
m
e i � ej
����� i �����
�∆j� 1
e i � e j
����� � g ij � 1
�����
In the series of formulae below we have repeated the formulae derived for the exterior product
and shown its equivalent for the regressive product in the same box for comparison. We have
used the fact that ∆ j i and g m ij
are symmetric to interchange the indices.
ei
m
e i
m
ei
m
e i
m
� ej �����
j �����
�∆i�
1
m
����� i �����
� ej �∆j� 1
m
����� m
� ej � gij� 1
m
�����
����� m
� g ij
� 1
�����
� ej
m
The complement of a simple element is simple
ei
m
e i
m
ei
m
e i
m
� ej �����
j
�∆i
m
����� i
� ej �∆j
m
����� m
� ej � gij m
����� m
� ej � g
m
ij
The expression of the complement of a basis element in terms of its cobasis element in the
reciprocal basis allows us a straightforward proof that the complement of a simple element is
simple.
To show this, suppose a simple element Α m expressed in terms of its factors Αi .
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