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Paper   IPM / P / 7143
School of Physics
  Title:   Dirac Structures on Modules 1
  Author(s):  A. Shafei Deh Abad
  Status:   Preprint
  Journal:
  No.:  0
  Year:  1998
  Supported by:  IPM
  Abstract:
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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