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A.A. Kirillov introduced the family algebras in 2000. In this paper we studythe noncommutative Poisson bracket P on the classical family algebra. We showthat P is the first-order deformation from the classical family algebra to thequantum family algebra. We will prove that the noncommutative Poisson bracketis in fact a Hochschild 2-coboundary therefore the deformation isinfinitesimally trivial. In the last part of this paper we also talk aboutMackey's analogue and the quantization problem of the family algebras.

The noncommutative Poisson bracket and the deformation of the family algebras