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
0
6
Z
0
7 1
Z
0 1
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