$$\mathfrak{g}$$ -Relative Star Products

Abstract

We consider Kontsevich star products on the duals of Lie algebras. Such a star product is relative if, for any Lie algebra, its restriction to invariant polynomial functions is the usual pointwise product. Let \(\mathfrak{g}\) be a fixed Lie algebra. We shall say that a Kontsevich star product is \(\mathfrak{g}\)-relative if, on \(\mathfrak{g}\)*, its restriction to invariant polynomial functions is the usual pointwise product. We prove that, if \(\mathfrak{g}\) is a semi-simple Lie algebra, the only strict Kontsevich \(\mathfrak{g}\)-relative star products are the relative (for every Lie algebras) Kontsevich star products.