Grassmann Algebra

TheExteriorProduct.nb 21
Any of these three representations will be called the basis n-element of the space. ����� 1 may also
be described as the cobasis of unity. Note carefully that ����� 1 is different in different bases.
� Tables and palettes of cobasis elements
We can create tables and palettes of basis elements with their corresponding cobasis elements
by the commands CobasisTable and CobasisPalette.
Entering CobasisTable[] or CobasisPalette[] will generate a table or palette of the
basis elements of the algebra generated by the currently declared basis. For the default basis
�e1, e2, e3� we then have:
CobasisTable��
� BASIS COBASIS
�
0
1 i � j � k
�
l
i j � k
�
l
j ��i � k�
�
l
k i � j
�
2
i � j k
�
2
i � k �j
�
2
j � k i
�
3
i � j � k 1
If you want to generate a default basis table or palette for a space of another dimension, include
the dimension as the argument:
CobasisTable�2�
� BASIS COBASIS
� 0
� l
1 e1 � e2
e1
e2
�
l
e2 �e1
�
2
e1 � e2 1
If you want to create a table or palette for your own basis, first declare the basis.
2001 4 5
DeclareBasis���, �, ���
��, �, ��