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 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\
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