The case *p*=1 is of particular interest. Indeed, those cocycles can be used to construct several inequivalent cyclic cocycles on the super-circle and this, in turns, allows one to construct many inequivalent central extensions of graded loop algebras
(i.e. super Kac-Moody algebras)[3]. Even, in this simple case *p*=1, the total number of
independent cyclic one-cocycles grows rapidly with *N*. Previous formulae give and . Altogether In the cases *N*=1,2,3, for instance, one gets *d* =1, and
.
For example, in the case N=2, p=1, the most general current of degree 1 reads . Taking into account cyclic symmetry does not bring anything new in this simple case. Let us write
. Then . The condition reads

Therefore closed currents are such that , and . These three algebraic relations are independent, so we have 8-3=5 independent cocycles (closed currents). The most general one reads

Let us take, for instance, . Let , with
(and a similar notation for *b*).
The bilinear form on *Gr*(2) corresponding to is but so that Considering cyclic cocycles as bilinear forms on *Gr*(2) and using the notation introduced in eq. (3.12), one can write a basis of this space as follows.

For instance, evaluated on , gives as we already know. We refer to the article [3] for more details concerning the case *p*=1. Since the main subject of the present article is the study of the case *p*>1, we wish now to illustrate the previous techniques by studying an example with *p*=2, *N*=2.

Mon May 20 14:40:14 MET DST 1996