Grassmann Algebra

TheExteriorProduct.nb 16
� Generating bases of exterior linear spaces
The function Basis�[m] generates a list of basis elements of � arranged in standard order
m
from the declared basis of �. For example, we declare the basis to be that of a 3-dimensional
1
vector space, and then compute the bases of �, �, �, and �. 0 1 2 3
�3
�e1, e2, e3�
Basis��0�
1
Basis��1�
�e1, e2, e3�
Basis��2�
�e1 � e2, e1 � e3, e2 � e3�
Basis��3�
e1 � e2 � e3
Basis�[] generates a list of the basis elements of all the �. This is a basis of the three-
m
dimensional Grassmann algebra. As a linear space, the three-dimensional Grassmann algebra
has 23 = 8 dimensions corresponding to its 8 basis elements.
Basis���
�1, e1, e2, e3, e1 � e2, e1 � e3, e2 � e3, e1 � e2 � e3�
� Tables and palettes of basis elements
Creating a table of basis elements
If you would like to create a table of the basis elements of all the exterior linear spaces induced
by the declared basis, you can use the GrassmannAlgebra command BasisTable[]. For
example if you are working in a three-dimensional vector space and declare the basis to be
{i,j,k}:
DeclareBasis��i, j, k��
�i, j, k�
Entering BasisTable[] would then give you the 8 basis elements of the 3-dimensional
Grassmann algebra:
2001 4 5