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|Paper IPM / Physic / 7143||
In this paper we give a simple definition of Dirac structures on
modules and on vector bundles which includes the existing ones,
and complex structures on vector bundles, as special cases. Among
other thing we prove:
1) Each two Dirac structures on a (Hermitian) module M are (isometrically) isomorphic(Hermitian) modules. Moreover, the set of all Dirac structures on Mis in one-to-one correspondence with Aut(M).
2) Let M be a smooth manifold, and let η be a (Hermitian) vector bundle over M. Then, to each Dirac structure on η, there corresponds a unique Dirac structure on the (Hermitian) C∞(M)−module of its sections. Conversely, to each Dirac structure on a Hermitian finitely generated projective C∞(M)−module there corresponds a unique Dirac structure on the associated (Hermitian) vector bundle over M.
3) Let M be a Hilbert R-module. Then to each Dirac structure on Mand to each state of R there corresponds a unique Dirac structure on the associated Hilbert space.
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