Grassmann Algebra

TheComplement.nb 26
� Simplifying complements
The GrassmannAlgebra function for converting the complements in an expression to a simpler
form (wherever possible) is GrassmannSimplify (or its alias �). 'Simpler' here means that
the complement operation appears less frequently in the result. Remember that the complement
depends on the metric.
Example: Simplifying complements with a Euclidean metric
GrassmannSimplify will take any Grassmann expression and convert any complements of
basis elements in it. Suppose we are in a 3-space with a Euclidean metric.
�3; DeclareDefaultMetric��
��1, 0, 0�, �0, 1, 0�, �0, 0, 1��
�� e2
����� �������������
� e2 � e3
�����
�������������������� �����
�
e1 � e2
GrassmannSimplify works on lists and matrices of Grassmann expressions to convert any
complements occurring in any of the elements. Here is a matrix whose elements we have
complemented:
M � ��a e1, be2�, ��b e2, ce3��; MatrixForm� M
����� �
���������� ��������
� ae1 be2
����
�����
����������
� �b
��������
e2 ce3 �
Applying � to this matrix of complements simplifies each element.
Mc � �� M
����� �; MatrixForm�Mc�
� ae2 � e3 �b e1 � e3
be1 � e3 ce1 � e2
�
It also works on complements of complements of symbolic expressions. In a 3-space, the
complement of the complement of any element is the element itself.
��������� ���������
��Α � Β� m k
Α m � Β k
In a 4-space however, the sign of the result will depend on the grades involved.
2001 4 5
��������� ���������
�4; ��Α � Β� m k
��1� k�m Α m � Β k