We summarize here some properties of Grassmann algebras and introduce a few notations that we shall need in the sequel. More properties can be found in [7] or [4].

A Grassmann algebra *Gr*(*N*) can be defined as a (real or complex) associative unital algebra with *N*
generators satisfying the defining relations and therefore In the following we choose the
basic field as the field of complex numbers. As a vector space, it has dimension and a possible basis is where *I* is a multi-index with
together with the convention when *I*=0. is a graded vector space but also a graded commutative algebra: Calling the intrinsic grade of
*a* (i.e. 0 or 1 on even or odd product of resp.), we have It is sometimes useful to decompose *Gr*(*N*) as where denotes the space of nilpotents elements of *Gr*(*N*) and to notice that is an ideal in *Gr*(*N*). There is a canonical homomorphism
from *Gr*(*N*) to , i.e., a character, defined by and This homomorphism is actually a unique nontrivial character in *Gr*(*N*). An element
is invertible iff . In this case, calling one
gets .

The generating system used to define *Gr*(*N*) is by no means unique; it is clear that one can make a change of generating system by using an invertible matrix with scalar
coefficients but we should stress that one can also obtain new generating systems for the algebra
*Gr*(*N*) by choosing coefficients in *Gr*(*N*) itself rather than in . More precisely, if one calls a
chart (or frame) on *Gr*(*N*) a generating system such that the are odd and such that
, one proves that these two conditions imply Let then be a frame of *Gr*(*N*), the corresponding Berezin integral is the element of the dual given by

where In other words
this Berezin integral is equal to , the coefficient of *a* at the ``top
element'' . We have to stress the fact that the left hand side
of the previous equation is defined by the right hand side; in particular the symbol has nothing to do with a 1-form : see section 2.3.
If is
another chart of *Gr*(*N*) -that we obtain from by an invertible matrix with coefficients in , the corresponding Berezin integral is defined in the same way but we have the relation

where

The fact that both Berezin integrals associated with charts and are related in
this way illustrates another interesting property of Grassmann algebras, namely the fact
that the dual of *Gr*(*N*) is singly generated as a right module over *Gr*(*N*), in other
words, if is such that then generates , i.e.

Mon May 20 14:40:14 MET DST 1996