Generalized symplectic geometries and elliptic equations

3

Remark 1.5 Cp,q has a volume element u = t\ • • • ep • ex • • • tq. It satisfies

o 6(6+1) 6(6+1)

u;2 = (-1) * , u;* = (-1) 2 ^ (1.3)

where 8 = p — q. Thus u is a selfadjoint involution if 8 = 0, — 1 (7710c?4). Moreover

when 8 = — l(mod4) u /ies m t/ie center of CVA since

coei + ( - l ) 5 e

t

^ = wcj + ( - 1 ) ^ = 0 V i,j . (1.4)

This implies that multiplication by |(1 — uf) is an idempotent endomorphism of Cp,q

i.e. Cp,q is not simple. Thus in this case every Cp,q module H has decomposition

H * H+@H- (1.5)

into the ±1 eigenspaces of ft = J\ • • • Jv • C\ • • • Cq.

Definition 1.6 A Fredholm selfadjoint operator T in an infinite dimensional Hilbert

space is called essentially indefinite if its essential spectrum, contains both posi-

tive and negative elements. In particular a selfadjoint involution C on an infinite

dimensional Hilbert space is called essential if both ker(7 — C) and ker(7 -f C) are

infinite dimensional. An infinite dimensional Hilbert (p,q)-m,odule (H,p) is called

essential if

(i) either 8 ^ — l(modi);

(it) or 8 = — I (mod 4) and the subspaces H± in the decomposition (1.5) are both

infinite dimensional. Equivalently, this means that the involution H = Jj • • • Jp •

C\ • • - Cq is essential. In particular a (p,q) s-module with grading R is called essential

if

(1) either p — q ^ 0(mof 4);

(ii) or p — q = 0(raof 4) and the involution RJ\ • • • JvC\ • • • Cq is essential.

A finite dimensional (p,q)~ module is called essential if either (p-q) ^ — 1 (mod

4) or if (p-q) = —1 (mod 4) the involution 0 is nontrivial i.e. H ^ 1. One defines

essential finite dimensional gradings in a similar fashion.

The topological meaning of essentiality will be revealed in Lemma 3.5

Denote by Mv"q the set of isomorphism classes of (p, c/)-modules. The direct sum

of modules induces on Mp,q a structure of abelian monoid. The Grothendieck group

associated to Mp,q (cf [Kal]) is denoted by Rp,q. Analyzing the algebraic structure of

the Clifford algebras one deduces the following periodicity result (see [Kal]).

Proposition 1.7 If p — q = p' — a' (mod 8) then

RP« £ Rv'-i\