The set of equivalent classes of invariant star products on \((G;\beta_ 1)\) (Q1377254)

This article, in conjunction with a previous one, proves Drinfeld's theorems about invariant star products, ISPS, on a connected Lie group \({\mathbf{G}}\) endowed with an invariant symplectic structure \(\beta_1 \in {\mathcal Z}^2({\mathfrak g})\). In particular, the authors prove that every formal \(2\)-cocycle \(\beta_{\hbar}\in\beta_1 + \hbar.{\mathcal Z}^2(\mathfrak g)[[\hbar]]\) determines an ISP, \(F^{\beta_{\hbar}}\), and conversely any ISP, \(F\), determines a formal \(2\)-cocycle \(\omega_{\hbar} \in \beta_1 + \hbar.{\mathcal Z}^2(\mathfrak g)[[\hbar]]\) such that \(F\) is equivalent to \(F^{\omega_{\hbar}}\). The authors also prove that two ISPS \(F^{\beta_{\hbar}}\) and \(F^{\omega_{\hbar}}\) are equivalent if and only if the cohomology classes \(F^{\beta_{\hbar}}\) and \(F^{\omega_{\hbar}}\) coincide. These properties define a bijection between the set of equivalent classes of ISP on \(({\mathbf{G}};\beta_1)\) and the set \(\beta_{\hbar}\in\beta_1 + \hbar.{\mathcal Z}^2(\mathfrak g)[[\hbar]]\).