Grassmann Algebra

TheComplement.nb 9
propagate through the algebra, and in consequence maintain a certain simplicity of structure in
the resulting formulae, or we can reformulate our approach to allow 1
���� ����
s � ��1� �1, where the
parity of s is even if �gij� is positive, and odd if it is negative. This latter approach would add
a factor ��1� s to many of the axioms and formulae developed from this chapter on. Because
of the introductory nature of the book however, we have chosen to adopt the former approach,
allowing readers less versed in the algebra to be undistracted by the extra complexities of
pseudo-Riemannian metrics.
From this point on then, we take the value of � to be the reciprocal of the positive square root
of the determinant of the coefficients of the complement mapping gij .
� �
1
��������������������
���������������
�gij�
This relation clearly also implies that the array �gij� is required to be non-singular.
5.4 The Euclidean Complement
Tabulating Euclidean complements of basis elements
The Euclidean complement of a basis m-element may be defined as its cobasis element.
Conceptually this is the simplest correspondence we can define between basis m-elements and
basis (nÐ1)-elements. In this case the matrix of the metric tensor is the identity matrix, with the
result that the 'volume' � of the basis n-element is unity. We tabulate the basis-complement
pairs for spaces of two, three and four dimensions.
Basis elements and their Euclidean complements in 2-space
2001 4 5
� BASIS COMPLEMENT
�
0
1 e1 � e2
� l
e1
e2
�
l
e2 �e1
�
2
e1 � e2 1
5.14