Grassmann Algebra

TheComplement.nb 21
Verifying the symmetry of the induced metrics
It is easy to verify the symmetry of the induced metrics in any particular case by inspection.
However to automate this we need only ask Mathematica to compare the metric to its transpose.
For example we can verify the symmetry of the metric induced on � by a general metric in 4-
3
space.
�4; DeclareMetric���; M3 � Metric��3�; M3 � Transpose�M3�
True
� The metric for a cobasis
In the sections above we have shown how to calculate the metric induced on � m by the metric
defined on �. Because of the way in which the cobasis elements of � are naturally ordered
1 m
alphanumerically, their ordering and signs will not generally correspond to that of the basis
elements of � . The arrangement of the elements of the metric tensor for the cobasis elements
n�m
of � will therefore differ from the arrangement of the elements of the metric tensor for the basis
m
elements of � . The metric tensor of a cobasis is the tensor of cofactors of the metric tensor of
n�m
the basis. Hence we can obtain the tensor of cofactors of the elements of the metric tensor of �
m
by reordering and resigning the elements of the metric tensor induced on � .
n�m
As an example, take the general metric in 3-space discussed in the previous section. We will
show that we can obtain the tensor of the cofactors of the elements of the metric tensor of � by
1
reordering and resigning the elements of the metric tensor induced on �. 2
The metric on � 1 is given as a correspondence G1 between the basis elements and cobasis
elements of � 1 .
��������
�
e1
e2
��������
� e2 � e3
�������
��������
� ��G1� ��e1 � e3�
� � e1 � e2 �
; G1
���������
�
�
�1,1 �1,2 �1,3
�1,2 �2,2 �2,3
e3
�1,3 �2,3 �3,3
The metric on � is given as a correspondence G2 between the basis elements and cobasis
2
elements of � 2 .
2001 4 5
��������
�
G2 �
e1 � e2
e1 � e3
e2 � e3
�����������
�
��������
� e3 �������
��������
� � G2� �e2
� � e1 �
;
��2 1,2 � �1,1 �2,2 ��1,2 �1,3 � �1,1 �2,3 ��1,3 �2,2 � �1,2 �2,3
��1,2 �1,3 � �1,1 �2,3
��2 1,3 � �1,1 �3,3 ��1,3 �2,3 � �1,2 �3,3
��1,3 �2,2 � �1,2 �2,3 ��1,3 �2,3 � �1,2 �3,3 ��2 2,3 � �2,2 �3,3
���������
�
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