\(\mathfrak g\)-relative star products (Q5944704)

This paper studies Kontsevich star products on duals of Lie algebras. Each such star product, associated to a linear Poisson structure \(\alpha\), is given by a universal integral formula \[ (u_1*_\alpha U_2)(\xi)= \int_{{\mathfrak g} \times{\mathfrak g}} \widehat u_1(X)\widehat u_2(Y){F(X) F(Y)\over F(X \times_\alpha Y)}e^{2\pi i\langle X\times_\alpha Y,\xi \rangle} dX dY, \] where \(u_1,u_2\) are polynomial functions on \({\mathfrak g}^*\), \(X\times_\alpha Y\) is the Baker-Campbell-Hausdorff formula in \(X,Y\) and \(F\) is a formal series of the form \[ 1+\sum^\infty_{n=1} \sum_{s_1,\dots,s_p, |s|=2n}a_{s_1\dots s_p} \text{Tr(ad} X)^{s_1} \cdots\text{Tr(ad} X)^{s_p}. \] The product is called a strict Kontsevich \(*\)-product if there exists a function \(f\), holomorphic in a neighborhood of 0, such that \[ f(0)=1,\quad\text{and}\quad F(X)=\det \bigl(f( \text{ad} X)\bigr). \] A Kontsevich star product is called relative if its restriction to the space of invariant polynomial functions is the usual pointwise product. If \(\alpha\to *_\alpha\) is a Kontsevich star product then \(*_\alpha\) is called \({\mathfrak g}\)-relative if \(u_1*_\alpha u_2=u_1u_2\) for the canonical Poisson tensor \(\alpha\) on \({\mathfrak g}^*\) and any invariant polynomial functions \(u_1,u_2\) on \({\mathfrak g}\). The main result of the paper shows that if \({\mathfrak g}\) is a semisimple Lie algebra, the only strict Kontsevich \({\mathfrak g}\)-relative star products are the relative (for every Lie algebra) Kontsevich star products. The paper concludes with an example of an entire function that gives rise to a relative strict Kontsevich star product.