Grassmann Algebra

ExploringCliffordAlgebra.nb 42
C9 � C8 �. �e1 � �, e2 � �, e1 � e2 � ��; PaletteForm�C9�
1 � � �
� �1 � ��
� �� �1 �
� � �� �1
Having verified that the structure is indeed quaternionic, let us return to the original
specification in terms of the basis of the Grassmann algebra. A quaternion can be written in
terms of these basis elements as:
Q � a � be 1 � ce 2 � de 1 � e 2 � �a � be 1 � � �c � de 1 ��e 2
Now, because e1 and e2 are orthogonal, e1 � e2 is equal to e1 � e2 . But for any further
calculations we will need to use the Clifford product form. Hence we write
Q � a � be1 � ce2 � de1 � e2 �
�a � be1� � �c � de1� � e2
12.66
Hence under one interpretation, each of e 1 and e 2 and their Clifford product e 1 � e 2 behaves as
a different imaginary unit. Under the second interpretation, a quaternion is a complex number
with imaginary unit e2 , whose components are complex numbers based on e1 as the imaginary
unit.
12.15 Clifford Algebras of a 3-Space
� The Clifford product table in 3-space
In this section we explore the Clifford algebras of 3-space. First we declare a (not necessarily
orthogonal) basis for the 3-space, and generate the associated Clifford product table. Because of
the size of the table, only the first few columns are shown in the print version.
2001 4 26
�3; C1 � CliffordProductTable��; PaletteForm�C1 �
1 � 1 1 � e1 1 � e2 1 � e3
e1 � 1 e1 � e1 e1 � e2 e1 � e3
e2 � 1 e2 � e1 e2 � e2 e2 � e3
e3 � 1 e3 � e1 e3 � e2 e3 � e3
�e1 � e2 � � 1 �e1 � e2 � � e1 �e1 � e2 � � e2 �e1 � e2 � � e3 �e1
�e1 � e3 � � 1 �e1 � e3 � � e1 �e1 � e3 � � e2 �e1 � e3 � � e3 �e1
�e2 � e3 � � 1 �e2 � e3 � � e1 �e2 � e3 � � e2 �e2 � e3 � � e3 �e2
�e1 � e2 � e3� � 1 �e1 � e2 � e3 � � e1 �e1 � e2 � e3 � � e2 �e1 � e2 � e3 � � e3 �e1 �