Grassmann Algebra

ExploringGrassmannAlgebra.nb 27
Z � GrassmannPower�X, �Π�
Ξ 0 �Π �Πe1 Ξ 0 �1�Π Ξ1 �Πe2 Ξ 0 �1�Π Ξ2 �ΠΞ 0 �1�Π Ξ3 e1 � e2
We verify that these are indeed inverses.
���Y � Z, Z � Y��
�1, 1�
9.6 Solving Equations
� Solving for unknown coefficients
Suppose we have a Grassmann number whose coefficients are functions of a number of
parameters, and we wish to find the values of the parameters which will make the number zero.
For example:
Z � a � b � 1 � �b � 3 � 5�a� e1 � �d � 2�c � 6� e3 � �e � 4�a��e1 � e2 �
�c � 2 � 2�g� e2 � e3 � ��1 � e � 7�g � b� e1 � e2 � e3;
We can calculate the values of the parameters by using the GrassmannAlgebra function
GrassmannScalarSolve.
GrassmannScalarSolve�Z � 0�
��a � 1
���� ,b��
2 1 ���� ,c��1, d ��8, e ��2, g ��
2 1
���� ��
2
GrassmannScalarSolve will also work for several equations and for general symbols
(other than basis elements).
Z1 � �a ��x� � by���2�x� � f �� cx� dx� y,
�7 � a��y �� �13 � d��y � x � f�;
GrassmannScalarSolve�Z1�
��a � 7, c ��7, d � 13, b � 13
������� ,f� 0��
2
GrassmannScalarSolve can be used in several other ways. As with other Mathematica
functions its usage statement can be obtained by entering ?GrassmannScalarSolve.
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