Grassmann Algebra

ExploringHypercomplexAlgebra.nb 18
Body�Α m ����Α m � ��Α m ���� Α m
The soul of a symmetrized sum
The soul of Α m ���� Α m is then the non-scalar residue given by the formula:
m�2
Soul�Α ����Α � � � ��1 � ��1�
m m �m�Λ�� Σ
Λ�0
m,Λ,m
Α1�����
�Α2
m Λ m
(Here we have used the fact that because of the coefficient �1 � ��1� �m�� �, the
terms with Λ equal to m–1 is always zero, and we only sum to m–2).
11.38
11.39
For convenience in generating examples of this expression, we temporarily define the
soul of Α���� Α as a function of m, and denote it Η[m]. To make it easier to read we
m m
convert the generalized products where possible to exterior and interior products.
m�2
Η�m_� :� � ��1 � ��1��m�Λ�� Σ
Λ�0
GeneralizedProductSimplify
m,Λ,m
Α1�����
�Α2
m Λ m
With this function we can make a palette of souls for various values of m. We do this
below for m equals 1 to 6 to cover spaces of dimension from 1 to 6.
PaletteTemplate�Table��m, �m��, �m, 1, 6���
1 0
2,0,2
2 2 Σ Α1 � Α2
2 2
3 2 �Α1
3
4 2 �Α1
4
5 2 �Α1
5
6 2 �Α1
6
3,1,3
� Α2�
Σ
1 3
4,2,4 4,0,4
� Α2�
Σ � 2 Σ Α1 � Α2
2 4
4 4
5,1,5
� Α2�
Σ � 2 �Α1
1 5
5
6,2,6
� Α2�
Σ � 2 �Α1
2 6
6
��
5,3,5
� Α2�
Σ
3 5
6,4,6 6,0,6
� Α2�
Σ � 2 Σ Α1 � Α2
4 6
6 6
Of particular note is the soul of the symmetrized sum of two different simple 2elements.
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