Grassmann Algebra

TheRegressiveProduct.nb 6
� �8: There is a unit n-element which acts as an identity under the regressive product.
�������1 n ,1 n �� n � : Α m � 1 n � Α m
� �9: Non-zero n-elements have inverses with respect to the regressive product.
������� 1
���� ,
Αn
1
����
Αn
��� :1 � Α� n n n 1
���� , �������Α, Α � 0 ��� n n n n
� �10: The regressive product of elements of odd complementary grade is anticommutative.
Α m � Β k
Αn
� ��1� �n�m���n� k�  k
� �11: Additive identities act as multiplicative zero elements under the regressive
product.
� Α m
�������0 k ,0 k �� k � :0 k � Α m �� 0
k�m�n
� �12: The regressive product is both left and right distributive under addition.
�Α m �Β m
Α m ��Β r
��à r
� Α m � Γ r
�Β� Γ
m r
�Γ� � Α�Β �Α� Γ
r m r m r
� �13: Scalar multiplication commutes with the regressive product.
2001 4 5
Α ��a�Β� � �a�Α��Β � a��Α � Β� m k m k m k
3.4
3.5
3.6
3.7
3.8
3.9