# An algebra of deformation quantization for star-exponentials on complex symplectic manifolds

Abstract : The cotangent bundle $T^*X$ to a complex manifold $X$ is classically endowed with the sheaf of $\cor$-algebras $\W[T^*X]$ of deformation quantization, where $\cor\eqdot \W[\rmptt]$ is a subfield of $\C[[\hbar,\opb{\hbar}]$. Here, we construct a new sheaf of $\cor$-algebras $\TW[T^*X]$ which contains $\W[T^*X]$ as a subalgebra and an extra central parameter $t$. We give the symbol calculus for this algebra and prove that quantized symplectic transformations operate on it. If $P$ is any section of order zero of $\W[T^*X]$, we show that $\exp(t\opb{\hbar} P)$ is well defined in $\TW[T^*X]$.
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https://hal.archives-ouvertes.fr/hal-00084660
Contributor : Giuseppe Dito <>
Submitted on : Monday, July 10, 2006 - 3:27:58 AM
Last modification on : Tuesday, May 14, 2019 - 11:08:26 AM
Long-term archiving on : Monday, April 5, 2010 - 11:57:45 PM

### Citation

Giuseppe Dito, Pierre Schapira. An algebra of deformation quantization for star-exponentials on complex symplectic manifolds. 2006. ⟨hal-00084660⟩

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