Grassmann Algebra

ExploringCliffordAlgebra.nb 37
a � b � a � b � a ���� b � a����� 0 �b � ab
Thus the real algebra is isomorphic with the Clifford algebra of a space of zero dimensions.
� Clifford algebras of a 1-space
12.65
We begin our discussion of Clifford algebras with the simplest case: the Clifford algebras of 1space.
Suppose the basis for the 1-space is e1 , then the basis for the associated Grassmann
algebra is �1, e1 �.
�1; Basis���
�1, e1 �
There are only four possible Clifford products of these basis elements. We can construct a table
of these products by using the GrassmannAlgebra function CliffordProductTable.
CliffordProductTable��
��1 � 1, 1 � e1 �, �e1 � 1, e1 � e1 ��
Usually however, to make the products easier to read and use, we will display them in the form
of a palette using the GrassmannAlgebra function PaletteForm. (We can click on the palette
to enter any of its expressions into the notebook).
C1 � CliffordProductTable��; PaletteForm�C1 �
1 � 1 1 � e1
e1 � 1 e1 � e1
In the general case any Clifford product may be expressed in terms of exterior and interior
products. We can see this by applying ToInteriorProducts to the table (although only
interior (here scalar) products result from this simple case),
C2 � ToInteriorProducts�C1 �; PaletteForm�C2 �
1 e1
e1 e1 � e1
Different Clifford algebras may be generated depending on the metric chosen for the space. In
this example we can see that the types of Clifford algebra which we can generate in a 1-space
are dependent only on the choice of a single scalar value for the scalar product e1 ���� e1 . The
Clifford product of two general elements of the algebra is
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