G-systems and deformation of G-actions on ${\mathbb {R}}^{d}$Rd

Abstract

Given a (smooth) action φ of a Lie group G on \documentclass[12pt]{minimal}\begin{document}${\mathbb {R}}^d$\end{document}Rd we construct a differential graded algebra whose Maurer–Cartan elements are in one-to-one correspondence with some class of deformations of the (induced) G-action on \documentclass[12pt]{minimal}\begin{document}$C^{\infty }({\mathbb {R}}^d)[[\hbar ]]$\end{document}C∞(Rd)[[ℏ]]. In the final part of this note we discuss the cohomological obstructions to the existence and to the uniqueness (in a sense to be clarified) of such deformations.