Grassmann Algebra

TheRegressiveProduct.nb 38
SimpleQ�X�
a3 a5 � a2 a6
������������������������������
a1
a4 a5 � a2 a7
������������������������������
a1
� a8 �� 0&&
� a9 �� 0&& a4 a6 � a3 a7
������������������������������
a1
� a10 �� 0
In this case of a 2-element in a 5-space, we have three conditions on the coefficients to satisfy
before being able to assert that the element is simple.
3.9 Product Formulae for Regressive Products
The Product Formula
The Common Factor Theorem forms the basis for many of the important formulae relating the
various products of the Grassmann algebra. In this section, some of the more basic formulae
which involve just the exterior and regressive products will be developed. These in turn will be
shown in Chapter 6: The Interior Product to have their counterparts in terms of exterior and
interior products. The formulae are usually directed at obtaining alternative expressions (or
expansions) for an element, or for products of elements.
The first formula to be developed (which forms a basis for much of the rest) is an expansion for
the regressive product of an (nÐ1)-element with the exterior product of two arbitrary elements.
We call this (and its dual) the Product Formula.
Let x
n�1 be an (nÐ1)-element, then:
�Α � Β�� x � �Α � x ��Β � ��1�
m k n�1 m n�1 k
m �Α ��Β� x �
m k n�1
To prove this, suppose initially that Α and Β
m k
Α �Α1� � � Αm and Β
m k
are simple and can be expressed as
�Β1 � � � Βk . Applying the Common Factor Theorem gives
�Α � Β�� x � ��Α1 � � � Αm���Β1 � � � Βk�� � x
m k n�1 n�1
m
i�1
k
j�1
� � ���1� i�1 ��Αi � x
� � ���1�m�j�1 ��Βj � x
n�1 ����Α1 � � � ���� i � � � Αm ���Β1 � � � Βk��
n�1 ����Α1 � � � Αm ���Β1 � � � ���� j � � � Βk��
� �Α � x ���Β1 � � � Βk� � ��1�
m n�1 m ��Α1 � � � Αm ���Β� x �
k n�1
Here we have used formula 3.13.
This result may be extended in a straightforward manner to the case where Α m and Β k
are not
simple: since a non-simple element may be expressed as the sum of simple terms, and the
formula is valid for each term, then by addition it can be shown to be valid for the sum.
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