Grassmann Algebra

ExploringGrassmannAlgebra.nb 6
�3; �EvenGrade�Z�, OddGrade�Z��
�x � y � 2 �x � z� � z � y � x � 2x� y, �4 x� x � y � z�
EvenGrade and OddGrade are Listable.
To test whether a number is even or odd we use EvenGradeQ and OddGradeQ. Consider
again the general Grassmann number X in 3-space.
�EvenGradeQ�X�, OddGradeQ�X�,
EvenGradeQ�EvenGrade�X��, OddGradeQ�EvenGrade�X���
�False, False, True, False�
EvenGradeQ and OddGradeQ are not Listable. When applied to a list of elements, they
require all the elements in the list to conform in order to return True. They can of course still
be mapped over a list of elements to question the evenness or oddness of the individual
components.
EvenGradeQ �� �1, x, x � y, x � y � z�
�True, False, True, False�
Finally, there is the question as to whether the Grassmann number 0 is even or odd. It may be
recalled from Chapter 2 that the single symbol 0 is actually a shorthand for the zero element of
any of the exterior linear spaces, and hence is of indeterminate grade. Entering Grade[0] will
return a flag Grade0 which can be manipulated as appropriate. Therefore, both
EvenGradeQ[0] and OddGradeQ[0] will return False.
�Grade�0�, EvenGradeQ�0�, OddGradeQ�0��
�Grade0, False, False�
� The grades of a Grassmann number
The GrassmannAlgebra function Grade will determine the grades of each of the components
in a Grassmann number, and return a list of the grades. For example, applying Grade to a
general number X in 3-space shows that it contains components of all grades up to 3.
X
Ξ0 � e1 Ξ1 � e2 Ξ2 � e3 Ξ3 �Ξ4 e1 � e2 �
Ξ5 e1 � e3 �Ξ6 e2 � e3 �Ξ7 e1 � e2 � e3
Grade�X�
�0, 1, 2, 3�
Grade will also work with more general Grassmann number expressions. For example we can
apply it to the number Z which we previously defined above.
2001 4 5
Z
x � y � x ��2 � y� � ��2 � z� � z � y � x