On a formula of Coll-Gerstenhaber-Giaquinto (Q1299354)

The authors study formal deformation theory of module algebras of a given bialgebra \(B\). In particular they find a sufficient condition for a deformation of the coproduct in \(B\) to determine deformations of all \(B\)-module algebras. As an application the authors give new proofs of the following results stated for a ring \(K\) containing the field of rational numbers, a \(K\)-algebra \(A\) with product \(\mu_A\) and the deformation parameter \(h\). Theorem [\textit{J.~E.~Moyal}, Proc. Camb. Philos. Soc. 45, 99-124 (1949; Zbl 0031.33601); \textit{J.~Vey}, Comment. Math. Helv. 50, 421-454 (1975; Zbl 0351.53029)]. If the Abelian Lie algebra \(\mathcal G\) acts on \(A\) by derivations then for any element \(S\in{\mathcal G}\otimes{\mathcal G}\) the composition \(\mu_A\circ S\) is a 2-cocyle and the multiplication \(\mu_h=\mu_A\circ e^{hS}\) is associative. Theorem [\textit{V.~Coll, M.~Gerstenhaber} and \textit{A.~Giaquinto}, in Ring Theory 1989, Isr. Math. Conf. Proc. 1, 396-403 (1989; Zbl 0684.16016)]. If the 2-dimensional Lie algebra \(\mathcal G\) with generators \(E,D\) and the commutator relation \([E,D]=E\) acts on \(A\) by derivations then for \(S=E\otimes D\) the composition \(\mu_A\circ S\) is a 2-cocyle and the multiplication \(\mu_h=\mu_A\circ(1+hE\otimes 1)^{1\otimes D}\) is associative.