4

L. I. Nicolaescu

We define in a similar way the monoid Mpq of isomorphism classes of finite di-

mensional (p, q) s-modules. Let Rp,q denote its associated Grothendieck group. If

H = H+ © //_ is a

Cp,q

supermodule than Cg+i = ProjH+ — Projn_ is a selfad-

joint involution on H anticommuting with the generators of the Cp'9-action (briefly

a (p,q)-grading) and thus extends the Cp,q structure on H to a Cp,q+1 structure.

Conversely, a (p, g)-grading on a (p, g)-module induces a super Cp,q structure so that

we have

Lemma 1.8

Rpq £ R™+1.

Using the periodicity result of Propositionl.7 one can replace the notation Rp,q

by Rp~q and similarly for the R's. The inclusion Cp,q -• Cp,q+1 induces a map

iPiq :

Rp'q+1

- k™.

Following [ABS] we introduce

/\p—q — Ap^ — COKer lp,q = It ' flp^qlt.

For any finite dimensional (p, q) s-module M with grading 7 7 we denote by [A/, 77] its

image in APiq. If (MA,.. 77^) (fc = 1,2) are two (p, /) s-modules then [Mi, 7/1] = [M2,7?2]

iff there exist (p, q -f 1) s-modules JVjt (A; = 1,2) such that

Mi 6 Ari = M2 0 N2 as (p, 4) s - modules.

The groups Ap,q will play an important role in this paper. We list bellow the

groups/?0,9 and Ao,q for q = 0,... , 7.

Table 1.1

1 q

#u'9

1

^ 0

) ?

0

z

z

1

zez

z2

2

z

z2

3

z

0

4

Z

Z

5

Z® Z

0

6

Z

0

7 1

Z

0 1