Grassmann Algebra

TheExteriorProduct.nb 22
CobasisPalette��
� BASIS COBASIS
� 0
� l
� l
� l
� 2
� 2
� 2
1 � � � � �
� � � �
� ��� � ��
� � � �
� � � �
� � � ��
� � � �
� 3 � � � � � 1
We can use the function Cobasis to compute the cobasis elements of any basis element or list
of basis elements from any of the � of the currently declared basis. For example, suppose you
m
are working in a space of 5 dimensions.
�5; Cobasis��e1, e1 � e3 � e4, e2 � e3��
�e2 � e3 � e4 � e5, e2 � e5, e1 � e4 � e5�
The cobasis of a cobasis
Let ei be a basis element and ei be its cobasis element. Then the cobasis element of this
m ����� m
cobasis element is denoted ej and is defined as expected by the product of the remaining basis
����� m
�����
elements such that:
ei
m
�����
� ej
����� m
�����
The left-hand side may be rearranged to give:
�∆ij�e1 � e2 � � � en
ei � ej � ��1�
����� m
����� m
�����
m��n�m� �ej � ei
����� m ����� m
�����
which, by comparison with the definition for the cobasis shows that:
2001 4 5
ej
m
����� �����
� ��1� m��n�m� �ei
m
2.26
2.27