Let \(A\) be an associative unitary algebra and let \(B\) be a bialgebra over a field of characteristic zero whose action on \(A\) is compatible with the multiplication \(A\!\otimes\! A\rightarrow A\) and the unit of \(A\). By definition (see [A. Giaquinto and J. J. Zhang, J. Pure Appl. Algebra 128, No. 2, 133–151 (1998; Zbl 0938.17015)]), a universal deformation formula is an element of an extension of \(B\!\otimes\! B\) that determines a deformation of \(A\) by twisting the multiplication. The authors remark that this construction firstly appears in quantum mechanics and physics (e.g. [H. J. Groenewold, Physica 12, 405–460 (1946; Zbl 0060.45002)]) for bialgebras which are the universal enveloping algebras of Lie algebras.
The aim of the paper under review is to describe in a didactic style the main ingredients of the modern theory and to discuss basic results about bialgebra actions and moduli spaces of universal deformation formulas. Among other things, the authors give a historical background and a brief survey of some classical results and examples concerning relations between deformations and universal deformation formulas, they also discuss some useful interpretations in the framework of the theory of operads, generalizations to other types of algebras and their diagrams, etc.