Grassmann Algebra

TheExteriorProduct.nb 14
when general Grassmann expressions, for example those involving Clifford or hypercomplex
numbers, are being considered. Clifford numbers have components of differing grade.
GrassmannAlgebra has the function Grade for calculating the grade of a Grassmann
expression. Grade works with single Grassmann expressions or with lists (to any level) of
expressions.
Grade�x � y � z�
3
Grade��1, x, x � y, x � y � z��
�0, 1, 2, 3�
It will also calculate the grades of the elements in a more general expression.
Grade�1 � x � x � y � x � y � z�
�0, 1, 2, 3�
Note that if we try to calculate the grade of an element which simplifies to 0, we will get the
flag Grade0 returned.
Grade�x � y � z � w�
Grade0
The expression x�y�z�w is zero because the current default basis is three-dimensional. In the
next section we will discuss bases and how to change them.
2.5 Bases
Bases for exterior linear spaces
Suppose e1 , e2 , �, en is a basis for � 1 . Then, as is well known, any element of � 1 may be
expressed as a linear combination of these basis elements.
A basis for � 2 may be constructed from the ei by taking all the essentially different non-zero
products ei1 � ei2 �i1, i2 :1,É,n�. Two products are essentially different if they do not
involve the same 1-elements. There are obviously � n
2
dimension of �. 2
n
� such products, making � � also the
2
In general, a basis for � m may be constructed from the ei by taking all the essentially different
products ei1 � ei2 � � � eim �i1, i2, É,im :1,É,n�. � m is thus � n
m �-dimensional.
Essentially different products are of course linearly independent.
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