Invariant Star Products of Wick Type: Classification and Quantum Momentum Mappings

Abstract

We extend our investigations on g-invariant Fedosov star products and quantum momentum mappings to star products of Wick type on pseudo-Kähler manifolds. Star products of Wick type can be completely characterized by a local description as given by Karabegov for star products with separation of variables. We separately treat the action of a Lie group G on C∞(M)[[ν]] by (pull-backs with) diffeomorphisms and the action of a Lie algebra g on C∞ (M)[[ν]] by (Lie derivatives with respect to) vector fields. Within Karabegov's framework, we prove necessary and sufficient conditions for a given star product of Wick type to be invariant in the respective sense. Moreover, our results yield a complete classification of invariant star products of Wick type. We also prove a necessary and sufficient condition for (the Lie derivative with respect to) a vector field to be even a quasi-inner derivation of a given star product of Wick type. We then transfer our former results about quantum momentum mappings for g-invariant Fedosov star products to the case of invariant star products of Wick type.