The family algebras are introduced by Kirillov in 2000. In this paper, we study the noncommutative Poisson bracket P on the classical family algebra 𝒞τ(𝔤). We show that P controls the first-order 1-parameter formal deformation from 𝒞τ(𝔤) to 𝒬τ(𝔤) where the latter is the quantum family algebra. Moreover, we will prove that the noncommutative Poisson bracket is in fact a Hochschild 2-coboundary, and therefore, the deformation is infinitesimally trivial. In the last part of this paper, we discuss the relation between Mackey’s analogue and the quantization problem of the family algebras.
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