Quantization of poisson pairs: the R-matrix approach

Abstract

We suggest an approach to the quantization problem of two compatible Poisson brackets in the case when one of them is associated with a solution of the classical Yang-Baxter equation. We show that the quantization scheme (a Poisson bracket → an associate algebra, quantizing this bracket in the spirit of the Berezin-Lichnerovicz deformation quantization → its representation in a Hilbert space) has to be enlarged. We represent the deformation algebras, quantizing the "R-matrix" brackets, in a space with an S-symmetric pairing, where S is a solution of the corresponding quantum Yang-Baxter equation. An example of quantization of an "exotic" harmonic oscillator is discussed. © 1992.