Grassmann Algebra

ExploringHypercomplexAlgebra.nb 12
In terms of �, �, and �
��Q� � a 2 �Α����Α��Α � Β����� �Α � Β�
��Q� � a 2 �Α���� Α��Α � Β� ���� �Α � Β�
Q � a � b � � c � � d �
��Q� � a 2 � �b � � c ������ �b � � c �� � �d �� �����d ��
� a 2 � b 2 �� ������ � c 2 �� ������ � d 2 �� ������
� a 2 � b 2 �� ���� �� � c 2 �� ���� �� � d 2 �� ���� ��
� a 2 � b 2 � c 2 � d 2
The Cayley-Dickson algebra
11.23
11.24
If we set Α and Β to be orthogonal in the general hypercomplex product table 11.21, we
retrieve the multiplication table for the Cayley-Dickson algebra with 4 generators and 2
parameters.
1 Α Β Α � Β
Α ��Α ���� Α� Α � Β ��Α ���� Α��Β
Β �Α� Β ��Β ���� Β� �Β ���� Β��Α
Α � Β �Α ���� Α��Β ��Β ���� Β��Α � �Α ���� Α���Β ���� Β�
11.25
In the notation we have used, the generators are 1, Α, Β, and Α�Β. The parameters are
Α����Α and Β����Β.
11.6 The Norm of Grassmann number
The norm
The first property that �, �, �, and � possess in common is that of being normed.
The norm of a Grassmann number X is denoted ��X� and has been defined in 11.5 as
the hypercomplex product of X with its conjugate.
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��X� � X ����X c � �Xb � Xs� �����Xb � Xs� � Xb 2 � Xs ���� Xs