We consider Rieffel’s
deformation quantization and Kontsevich’s star product on the duals of Lie algebras
equipped with linear Poisson brackets. We give the equivalence operator between
these star products and compare properties of these two main examples of
deformation quantization. We show that Rieffel’s deformation can be obtained by
considering oriented graphs Γ. We also prove that Kontsevich’s star product provides
a deformation quantization by partial embeddings on the space Cc∞(g) of C∞
compactly supported functions on a general Lie algebra g and a strict deformation
quantization on the space 𝒮(g∗) of Schwartz functions on the dual g∗ of
a nilpotent Lie algebra g. Finally, we give the explicit formulae of these
two star products and we deduce the weights of graphs occurring in these
expressions.
