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
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
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
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.