Journal of Lie Theory 15 (2005), No. 2, 457--495
Copyright Heldermann Verlag 2005
Spinor Types in Infinite Dimensions
Esther Galina
FAMAF-CIEM, Ciudad Universitaria, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina
galina@mate.uncor.edu
Aroldo Kaplan
FAMAF-CIEM, Ciudad Universitaria, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina
kaplan@mate.uncor.edu
Linda Saal
FAMAF-CIEM, Ciudad Universitaria, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina
saal@mate.uncor.edu
[Abstract-pdf]
The Cartan-Dirac classification of spinors into types is generalized to infinite dimensions. The main conclusion is that, in the statistical interpretation where such spinors are functions on $\Bbb Z_2^\infty$, any real or quaternionic structure involves switching zeroes and ones. There results a maze of equivalence classes of each type. Some examples are shown in $L^2({\Bbb T})$. The classification of spinors leads to a parametrization of certain non-associative algebras introduced speculatively by Kaplansky.
Keywords: Spinors, representations of the CAR, division algebras.
MSC: 81R10; 15A66
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