Grassmann Algebra

ExploringCliffordAlgebra.nb 38
�a � be1 � � �c � de1 ���ToScalarProducts
ac� bd�e1 � e1 � � bce1 � ade1
It is clear to see that if we choose e 1 ���� e 1 ��1, we have an algebra isomorphic to the complex
algebra. The basis 1-element e1 then plays the role of the imaginary unit �. We can generate this
particular algebra immediately by declaring the metric, and then generating the product table.
DeclareBasis�����; DeclareMetric����1���;
CliffordProductTable�� �� ToScalarProducts �� ToMetricForm ��
PaletteForm
1 �
� �1
However, our main purpose in discussing this very simple example in so much detail is to
emphasize that even in this case, there are an infinite number of Clifford algebras on a 1-space
depending on the choice of the scalar value for e 1 ���� e 1 . The complex algebra, although it has
surely proven itself to be the most useful, is just one among many.
Finally, we note that all Clifford algebras possess the real algebra as their simplest even
subalgebra.
12.14 Clifford Algebras of a 2-Space
� The Clifford product table in 2-space
In this section we explore the Clifford algebras of 2-space. As might be expected, the Clifford
algebras of 2-space are significantly richer than those of 1-space. First we declare a (not
necessarily orthogonal) basis for the 2-space, and generate the associated Clifford product table.
�2; C1 � CliffordProductTable��; PaletteForm�C1 �
1 � 1 1 � e 1 1 � e 2 1 � �e 1 � e 2 �
e1 � 1 e1 � e1 e1 � e2 e1 � �e1 � e2 �
e2 � 1 e2 � e1 e2 � e2 e2 � �e1 � e2 �
�e1 � e2 � � 1 �e1 � e2 � � e1 �e1 � e2 � � e2 �e1 � e2 � � �e1 � e2 �
To see the way in which these Clifford products reduce to generalized Grassmann products we
can apply the GrassmannAlgebra function ToGeneralizedProducts.
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