Deformation quantization of symplectic vector fields
Haoyuan Gao
TL;DR
This work addresses quantizing infinitesimal symplectomorphisms within the Fedosov framework by lifting symplectic vector fields to derivations of the deformed algebra. The authors construct, for each symplectic vector field $X$, an inner-corrected derivation $\widehat{X} = \mathcal{L}_X + [u_X, \cdot]$ that preserves the Fedosov flat subalgebra and induces a star-derivation $\widetilde{X}$ with $\widetilde{X} = X + O(\hbar)$; they also establish a non-abelian $2$-cocycle $\tau$ on the Lie algebra of symplectic vector fields. Extending this to Lie algebra actions, they define a quantized cocycle $(\tilde{\rho},\tilde{\tau})$ and obtain a cocycle cross product deformation $C^{\infty}(M)[[\hbar]] _{\chi} \triangleleft \mathcal{U}(\mathfrak{g})$ of the crossed product, which reduces to the classical structure at $\hbar = 0$. The results unify Fedosov deformation quantization with deformation theory of Hopf algebra actions and pave the way for algebraic index-theoretic applications and generalized quantizations of symplectic derivations.
Abstract
In this paper, we study deformation quantization of symplectic vector fields à la Fedosov. We show that each symplectic vector field can be quantized to a derivation of the deformed star algebra. Moreover, we show that this quantization yields a non-abelian $2$-cocycle on the Lie algebra of symplectic vector fields with values in the deformed star algebra. Therefore, we can quantize any Lie algebra action by symplectic vector fields.