Grassmann Algebra

TheRegressiveProduct.nb 3
3.2 Duality
The notion of duality
In order to ensure that the regressive product is defined as an operation correctly dual to the
exterior product, we give the defining axiom set for the regressive product the same formal
symbolic structure as the axiom set for the exterior product. This may be accomplished by
replacing � by �, and replacing the grades of elements and spaces by their complementary
grades. The complementary grade of a grade m is defined in a linear space of n dimensions to be
nÐm.
� ��� Α m � Α
n�m
� m � �
n�m
Note that here we are undertaking the construction of a mathematical structure and thus there is
no specific mapping implied between individual elements at this stage. In Chapter 5: The
Complement we will introduce a mapping between the elements of � and � which will lead to
m n�m
the definition of complement and interior product.
For concreteness, we take some examples.
Examples: Obtaining the dual of an axiom
The dual of axiom �6
We begin with the exterior product axiom �6:
The exterior product of an m-element and a k-element is an (m+k)-element.
�Α m �� m , Β k
�� k � � Α m � Β k
� �
m�k
To form the dual of this axiom, replace � with �, and the grades of elements and spaces by their
complementary grades.
� Α
n�m � �
n�m , Β
n�k
� �
n�k � � Α
n�m � Β
n�k
� �
n��m�k�
Although this is indeed the dual of axiom �6, it is not necessary to display the grades of what
are arbitrary elements in the more complex form specifically involving the dimension n of the
space. It will be more convenient to display them as grades denoted by simple symbols like m
and k as were the grades of the elements of the original axiom. To effect this transformation
most expeditiously we first let m' = nÐm, k' = nÐk to get
2001 4 5
� Α m' �� m' , Β k'
�� k' � � Α m' � Β k'
� �
�m'�k'��n
3.1