Grassmann Algebra

TheRegressiveProduct.nb 34
n
�
i�1
n
ai�ei
����� � � i�2
� Example: Application to 4-space
�ai�e1 � a1�ei�, a1 � 0
We check the validity of the formula above in a 4-space. Without loss of generality, let a1 be
unity.
��e2 � e3 � e4 � a2�e1 � e3 � e4 � a3 e1 � e2 � e4 � a4�e1 � e2 � e3 �
��a2�e1 � e2���a3�e1 � e3���a4�e1 � e4��
True
� Obtaining a factorization of a simple m-element
For simple m-elements
For m-elements known to be simple we can use the GrassmannAlgebra function Factorize
to obtain a factorization. Note that the result obtained from Factorize will be only one
factorization out of the generally infinitely many possibilities. However, every factorization
may be obtained from any other by adding scalar multiples of each factor to the other factors.
3.40
We have shown above that every (nÐ1)-element in an n-space is simple. Applying Factorize
to a series of general (nÐ1)-elements in spaces of 4, 5, and 6 dimensions corroborates the
formula derived in the previous section.
In each case we first declare a basis of the requisite dimension by entering �n , and then create a
general (nÐ1)-element X with scalars ai using the GrassmannAlgebra function
CreateBasisForm.
� A 3-element in a 4-space:
�4; X� CreateBasisForm�3, a�
a1 e1 � e2 � e3 � a2 e1 � e2 � e4 � a3 e1 � e3 � e4 � a4 e2 � e3 � e4
Factorize�X�
�a1 e3 � a2 e4����a1 e2 � a3 e4���a1 e1 � a4 e4�
�������������������������������� �������������������������������� �������������������������������� �������������������
� A 4-element in a 5-space:
2001 4 5
a 1 2
�5; X� CreateBasisForm�4, a�
a1 e1 � e2 � e3 � e4 � a2 e1 � e2 � e3 � e5 �
a3 e1 � e2 � e4 � e5 � a4 e1 � e3 � e4 � e5 � a5 e2 � e3 � e4 � e5