Grassmann Algebra

ExploringGrassmannAlgebra.nb 33
We get a slightly different result for the right quotient which we verify by multiplying from the
right.
�x � y� �x
�y � xC1 � C2 x � y
����y � xC1 � C2 x � y��x�
x � y
Here is a slightly more complex example.
�3; �x � y � z� ��x � 2�y � 3�z � x � z�
z � 3
���� �
2 9C1
�����������
2
� 3C2 � 3C3
�����������
2 � � x � 1 ���� �
2 3C1
�����������
2
� C2 � C3
�������
2 � �
y ��1 � 3C1 � 2C2 � C3� � C1 x � y � C2 x � z � C3 y � z � C4 x � y � z
In some circumstances we may not want the most general result. Say, for example, we knew
that the result we wanted had to be a 1-element. We can use the GrassmannAlgebra function
ExtractGrade.
ExtractGrade�1���x � y � z�� �x � 2�y � 3�z � x � z��
z � 3
���� �
2 9C1
����������� � 3C2 �
2
3C3
����������� � �
2
x � 1
���� �
2 3C1 ����������� � C2 �
2
C3 ������� � � y ��1 � 3C1 � 2C2 � C3�
2
Should there be insufficient information to determine a result, the quotient will be returned
unchanged. For example.
�x � y� �z
�x � y� �z
9.8 Factorization of Grassmann Numbers
The non-uniqueness of factorization
Factorization of a Grassmann number will rarely be unique due to the nilpotency property of the
exterior product. However, using the capability of GrassmannScalarSolve to generate
general solutions with arbitrary constants we may be able to find a factorization in the form we
require. The GrassmannAlgebra function which implements the attempt to factorize is
GrassmannFactor.
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