A comparison between Rieffel’s and Kontsevich’s deformation quantizations for linear Poisson tensors

Abstract

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 G. We also prove that Kontsevich?s star product provides a deformation quantization by partial embeddings on the space Cc8(g) of C8 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.