Grassmann Algebra

ExploringCliffordAlgebra.nb 18
DeclareBasis�4, Α�; B� Basis���
�1, Α1 , Α2 , Α3 , Α4 , Α1 � Α2 , Α1 � Α3, Α1 � Α4 , Α2 � Α3 , Α2 � Α4 , Α3 � Α4 ,
Α 1 � Α 2 � Α 3 , Α 1 � Α 2 � Α 4 , Α 1 � Α 3 � Α 4 , Α 2 � Α 3 � Α 4 , Α 1 � Α 2 � Α 3 � Α 4 �
� 2. Calculate the cobasis with respect to Α m
A � Cobasis���
�Α 1 � Α 2 � Α 3 � Α 4 , Α 2 � Α 3 � Α 4 , ��Α 1 � Α 3 � Α 4 �,
Α1 � Α2 � Α4 , ��Α1 � Α2 � Α3 �, Α3 � Α4 , ��Α2 � Α4 �, Α2 � Α3 ,
Α 1 � Α4 , ��Α1 � Α3�, Α1 � Α2 , Α4, �Α3 , Α2 , �Α1 ,1�
� 3. Take the reverse of the cobasis
R � GrassmannReverse�A�
�Α 4 � Α 3 � Α 2 � Α 1 , Α 4 � Α 3 � Α 2 , ��Α 4 � Α 3 � Α 1 �,
Α4 � Α2 � Α1 , ��Α3 � Α2 � Α1 �, Α4 � Α3 , ��Α4 � Α2 �, Α3 � Α2 ,
Α4 � Α1 , ��Α3 � Α1�, Α2 � Α1 , Α4, �Α3 , Α2 , �Α1 ,1�
� 4. Take the interior product of each of these elements with Β k
S � Thread� ���� R�
k
�Β � Α4 � Α3 � Α2 � Α1 , Β � Α4 � Α3 � Α2 , Β � ��Α4 � Α3 � Α1 �, Β � Α4 � Α2 � Α1,
k
k
k
k
Β � ��Α3 � Α2 � Α1 �, Β � Α4 � Α3 , Β � ��Α4 � Α2 �, Β � Α3 � Α2 , Β � Α4 � Α1 ,
k
k
k
k
k
Β k
� ��Α3 � Α1 �, Β � Α2 � Α1 , Β
k
k
� Α4 , Β k
� �Α3 , Β � Α2, Β � �Α1 , Β � 1�
k
k
k
� 5. Take the exterior product of the elements of the two lists B and S
2001 4 26
T � Thread�B � S�
�1 ��Β
k
Α2 ��Β� ��Α4 � Α3 � Α1 ��, Α3 ��Β
k
k
� Α4 � Α3 � Α2 � Α1 �, Α1 ��Β� Α4 � Α3 � Α2 �,
k
Α1 � Α2 ��Β� Α4 � Α3 �, Α1 � Α3 ��Β
k
k
Α2 � Α3 ��Β� Α4 � Α1 �, Α2 � Α4 ��Β
k
k
Α1 � Α2 � Α3 ��Β� Α4 �, Α1 � Α2 � Α4 ��Β
k
k
Α2 � Α3 � Α4 ��Β k
� Α4 � Α2 � Α1 �, Α4 ��Β k
� ��Α4 � Α2 ��, Α1 � Α4 ��Β k
� ��Α3 � Α1 ��, Α3 � Α4 ��Β k
� �Α1�, Α1 � Α2 � Α3 � Α4 ��Β� 1��
k
� ��Α3 � Α2 � Α1 ��,
� Α3 � Α2 �,
� Α2 � Α1 �,
� �Α3�, Α1 � Α3 � Α4 ��Β� Α2 �,
k