Grassmann Algebra

ExploringCliffordAlgebra.nb 40
e1 ���� e1 and e2 ���� e2 .
Note however that allocation of scalar values a to e1 ���� e1 and b to e2 ���� e2 would lead to
essentially the same structure as allocating b to e 1 ���� e 1 and a to e 2 ���� e 2 .
In the rest of what follows however, we will restrict ourselves to metrics in which e1 ���� e1 ��1
and e2 ���� e2 ��1, whence there are three cases of interest.
�e1 ���� e1 ��1, e2 ���� e2 ��1�
�e 1 ���� e 1 ��1, e 2 ���� e 2 ��1�
�e1 ���� e1 ��1, e2 ���� e2 ��1�
As observed previously the case �e1 ���� e1 ��1, e2 ���� e2 ��1� is isomorphic to the case
�e1 ���� e1 ��1, e2 ���� e2 ��1�, so we do not need to consider it.
� Case 1: �e1 ���� e1 ��1, e2 ���� e2 ��1�
This is the standard case of a 2-space with an orthonormal basis. Making the replacements in the
table gives:
C6 � C5 �. �e1 ���� e1 ��1, e2 ���� e2 ��1�; PaletteForm�C6 �
1 e1 e2 e1 � e2
e1 1 e1 � e2 e2
e2 ��e1 � e2 � 1 �e1
e1 � e2 �e2 e1 �1
Inspecting this table for interesting structures or substructures, we note first that the even
subalgebra (that is, the algebra based on products of the even basis elements) is isomorphic to
the complex algebra. For our own explorations we can use the palette to construct a product
table for the subalgebra, or we can create a table using the GrassmannAlgebra function
TableTemplate, and edit it by deleting the middle rows and columns.
TableTemplate�C6 �
1 e1 e2 e1 � e2
e1 1 e1 � e2 e2
e2 ��e1 � e2 � 1 �e1
e1 � e2 �e2 e1 �1
1 e1 � e2
e1 � e2
�1
If we want to create a palette for the subalgebra, we have to edit the normal list (matrix) form
and then apply PaletteForm. For even subalgebras we can also apply the GrassmannAlgebra
function EvenCliffordProductTable which creates a Clifford product table from just the
basis elements of even grade. We then set the metric we want, convert the Clifford product to
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