Some remarks on \(\mathfrak g\)-variant Fedosov star products and quantum momentum mappings. (English) Zbl 1078.53100

P. Xu [Commun. Math. Phys. 197, No. 1, 167–197 (1998; Zbl 0939.37048)] studied the notion of quantum momentum mapping for \({\mathfrak g}\)-invariant star products and raised the question whether the existence of a quantum momentum mapping follows from the existence of a classical momentum mapping. This question is answered in the negative in the paper under review, and the method used here improves results of K. Hamachi [Rev. Math. Phys. 14, No. 6, 601–621 (2002; Zbl 1040.53097)] and S. Gutt [Star products and group actions, Bayrischzell Workshop, April 26–29 (2002)].
The first part of the present paper includes a brief review of B. V. Fedosov’s construction of star products on symplectic manifolds. Additionally, one describes the derivations of the corresponding star products, and then one compares the Fedosov derivations obtained from different data. In Section 3 one obtains necessary and sufficient conditions in order for the Lie derivative with respect to a symplectic vector field to be a derivation of a Fedosov star product (Theorem 3.8). This result is then used to investigate the invariance of the star product with respect to a Lie algebra action. One further obtains criteria for existence of quantum Hamiltonians, eventually leading to the negative answer to P. Xu’s aforementioned question.


53D55 Deformation quantization, star products
53D20 Momentum maps; symplectic reduction
Full Text: DOI arXiv


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