Grassmann Algebra

TheComplement.nb 22
We can transform the columns of basis elements in this equation into the form of the preceding
one with the transformation T which is simply determined as:
� 0 0 1
�������
��������
T � 0 �1 0
� 1 0 0�
;
��������
T�
�
e1 � e2
e1 � e3
e2 � e3
��������
�
�
� e2 � e3
�������
��������
� e3
�������
��������
��e1 � e3� ; T��e2
� e1 � e2 � � e1 �
�
��������
�
And since this transformation is its own inverse, we can also transform G2 to T G2�T. We now
expect this transformed array to be the array of cofactors of the metric tensor G1 . We can easily
check this in Mathematica by entering the predicate
e1
e2
e3
��������
�
;
Simplify�G1.�T.G2.T� � Det�G1��IdentityMatrix�3��
True
� Creating tables of induced metrics
You can create a table of all the induced metrics by declaring a metric and then entering
MetricTable.
2001 4 5
�3; M� ��1, 0, Ν�, �0, �1, 0�, �Ν, 0,�1��;
DeclareMetric�M�; MetricTable
� METRIC
� 0
� l
� 2
� 3
1
� 1
�������
0
� Ν
0
�1
0
Ν ��������
0
�1 �
� �1
��������
0
�
0
�1 �Ν
Ν
2 Ν 0
���������
0
1�
1 �Ν 2