Grassmann Algebra

TheComplement.nb 27
Example: Simplifying complements with any metric
GrassmannSimplify will convert complements of basis elements in any declared metric.
�3; G� ��1, 0, Ν�, �0, �1, 0�, �Ν, 0,�1��;
DeclareMetric�G�; MatrixForm�G�
� 1 0 Ν
�������
��������
0 �1 0
� Ν 0 �1 �
A � ��� 1
����� ,e1
����� ������������� ��������������������
,e1� e2,e1�
e2 � e3��
� e1 � e2 � e3
�������������������������� ���������������
1 �Ν2 , Ν e1 � e2
�������������������� ���������������
1 �Ν2 � e2 � e3
������������������� ���������������
1 �Ν
Here, � is equal to the inverse of ���������������
1 �Ν 2 .
2 ,
Ν e1 e3 ���������������
������������������� ��������������� � �������������������
1 �Ν2 ��������������� , 1 �Ν
1 �Ν2 2 �
Of course, converting a complement of a complement gives a result independent of the metric,
which we can verify by taking the complement of the previous result and applying � again.
GrassmannSimplify� A
����� �
�1, e1, e1 � e2, e1 � e2 � e3�
Note that it is only because the space is three-dimensional in this example that there are no sign
differences between any of the basis elements in the list and the complement of its complement.
� Creating tables and palettes of complements of basis elements
GrassmannAlgebra has a function ComplementTable for tabulating basis elements and their
complements in the currently declared space and the currently declared metric. Once the table is
generated you can edit it, or copy from it.
Without further explanation we show how the functions are applied in some commonly
occurring cases.
Euclidean metric
Here we repeat the construction of the complement table of Section 5.4. Tables of complements
for other bases are obtained by declaring the bases, then entering the command
ComplementTable.
2001 4 5
�2; DeclareDefaultMetric�� �� MatrixForm
�
1 0
0 1 �