Mathematics – Quantum Algebra
Scientific paper
1999-05-27
Mathematics
Quantum Algebra
LaTeX2e, 18 pages, title changed, one definition and one proposition added to section 5
Scientific paper
In this paper we explicitly construct local $\nu$-Euler derivations $\mathsf E_\alpha = \nu \partial_\nu + \Lie{\xi_\alpha} + \mathsf D_\alpha$, where the $\xi_\alpha$ are local, conformally symplectic vector fields and the $\mathsf D_\alpha$ are formal series of locally defined differential operators, for Fedosov star products on a symplectic manifold $(M,\omega)$ by means of which we are able to compute Deligne's characteristic class of these star products. We show that this class is given by $1/\nu[\omega]+1/\nu[\Omega]$ where $\Omega = \sum_{i=1}^\infty \nu^i \Omega_i$ is a formal series of closed two-forms on $M$ the cohomology class of which coincides with the one introduced by Fedosov to classify his star products. Moreover, our result implies that the normalisation condition used by Fedosov does not have any effect on the isomorphy class of the resulting star product. Finally we consider star products that have additional algebraic structures and compute the effect of these structures on the corresponding characteristic classes of these star products. Specifying the constituents of Fedosov's construction we obtain star products with these special properties. Finally we investigate equivalence transformations between such special star products and prove existence of equivalence transformations being compatible with the considered algebraic structures.
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