Grassmann Algebra

TheExteriorProduct.nb 10
� �1: The sum of m-elements is itself an m-element.
�Α m �� m , Β m
� �2: Addition of m-elements is associative.
�Α m �Β m
� �Γ m
�� m � � Α m �Β m
� Α m � �Β m
�� m
�Γ� m
� �3: m-elements have an additive identity (zero element).
� �4: m-elements have an additive inverse.
�������0 m ,0 m �� m � : Α m � 0 m �Α m
��������Α m , �Α m �� m � : 0 m � Α m ��Α m
� �5: Addition of m-elements is commutative.
Α m �Β m
� Β�Α m m
� �6: The exterior product of an m-element and a k-element is an (m+k)-element.
�Α m �� m , Β k
� �7: The exterior product is associative.
2001 4 5
�Α m � Β k
��à r
�� k � � Α m � Β k
� Α m ��Β k
� Γ �
r
� �
m�k
2.7
2.8
2.9
2.10
2.11
2.12
2.13