Grassmann Algebra

ExploringGrassmannAlgebra.nb 25
Y � GrassmannPower�X, 3�
Ξ 0 3 � 3e1 Ξ 0 2 Ξ1 � 3e2 Ξ 0 2 Ξ2 � 3 Ξ 0 2 Ξ3 e1 � e2
Z � GrassmannPower�X, �3�
1
�������
�����������������
3 Ξ0 � 3e1 Ξ1
Ξ4 0
� 3e2 Ξ2
�����������������
Ξ4 0
� 3 Ξ3 e1 � e2
���������������������������
Ξ4 0
As usual, we verify with GrassmannSimplify.
���Y � Z, Z � Y��
�1, 1�
General powers
GrassmannPower will also work with general elements that are not expressed in terms of
basis elements.
X � 1 � x � x � y � x � y � z;
�Y � GrassmannPower�X, 3�, Z� GrassmannPower�X, �3��
�1 � 3x� 3x� y, 1 � 3x� 3x� y�
���Y � Z, Z � Y��
�1, 1�
Symbolic powers
We take a general Grassmann number in 2-space.
�2; X� CreateGrassmannNumber�Ξ�
Ξ0 � e1 Ξ1 � e2 Ξ2 �Ξ3 e1 � e2
Y � GrassmannPower�X, n�
Ξ 0 n � ne1 Ξ 0 �1�n Ξ1 � ne2 Ξ 0 �1�n Ξ2 � n Ξ 0 �1�n Ξ3 e1 � e2
Z � GrassmannPower�X, �n�
Ξ 0 �n � ne1 Ξ 0 �1�n Ξ1 � ne2 Ξ 0 �1�n Ξ2 � n Ξ 0 �1�n Ξ3 e1 � e2
Before we verify that these are indeed inverses of each other, we need to declare the power n to
be a scalar.
2001 4 5
DeclareExtraScalars�n�;
���Y � Z, Z � Y��
�1, 1�