Grassmann Algebra

TheExteriorProduct.nb 36
��Α 2 � Α 2 ���Simplify
2 �a3 a4 � a2 a5 � a1 a6� e1 � e2 � e3 � e4
Thus the condition that a 2-element in a 4-dimensional space be simple is that:
a3 a4 � a2 a5 � a1 a6 � 0
� Conditions for simplicity of a 2-element in a 5-space
The situation in 5-space is a little more complex.
�5; Α 2 � CreateBasisForm�2, a�
a1 e1 � e2 � a2 e1 � e3 � a3 e1 � e4 � a4 e1 � e5 � a5 e2 � e3 �
a6 e2 � e4 � a7 e2 � e5 � a8 e3 � e4 � a9 e3 � e5 � a10 e4 � e5
��Α 2 � Α 2 ���Simplify
2 ��a3 a5 � a2 a6 � a1 a8� e1 � e2 � e3 � e4 �
�a4 a5 � a2 a7 � a1 a9� e1 � e2 � e3 � e5 �
�a4 a6 � a3 a7 � a1 a10� e1 � e2 � e4 � e5 �
�a4 a8 � a3 a9 � a2 a10� e1 � e3 � e4 � e5 �
�a7 a8 � a6 a9 � a5 a10� e2 � e3 � e4 � e5�
2.34
For Α to be simple we require this product to be zero, hence each of the five coefficients must
2
be zero simultaneously. We can collect these five equations into a list A and obtain a solution to
them by using Mathematica's inbuilt Solve function:
A � �a3 a5 � a2 a6 � a1 a8 � 0,
a4 a5 � a2 a7 � a1 a9 � 0, a4 a6 � a3 a7 � a1 a10 � 0,
a4 a8 � a3 a9 � a2 a10 � 0, a7 a8 � a6 a9 � a5 a10 � 0�;
S � Solve�A, �a8, a9, a10��
������������������������������ ,a9 �� a4 a5 � a2 a7
������������������������������
��a8 �� a3 a5 � a2 a6
a1
a1
������������������������������ ��
,a10 �� a4 a6 � a3 a7
a1
By inspection we see that these three solutions correspond to the first three equations.
Consequently, from the generality of the expression for the 2-element we expect any three
equations to suffice.
In summary we can say that a 2-element Α in a 5-space is simple if any three of the following
2
equations is satisfied:
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