An analogue of the Livernet–Loday operad for two compatible brackets, which is a flat deformation of the bi-Hamiltonian operad is constructed. The Liver net–Lo day operad can be used to define ?-products and deformation quantization for Poisson structures. The constructed operad is used in the same way, introducing a definition of operadic deformation quantization of compatible Poisson structures.
Some constructions of deformation quantization are known now for the case which was the most important for, namely the algebra of functions on a smooth Poisson manifold; see, for example, the work of Kontsevich. It seems to be much more difficult to deal with deformation quantization of two compatible Poisson brackets – even on the level of introducing the problem and giving the necessary definitions.
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