Everybody knows what a normal Gabi-algebra is (Q6592210)

In this paper the authors study algebras \(A\) over a commutative ring \(k\) for which the forgetful functor \(\omega\) from the category \({}_A\mathcal M\) of left \(A\)-modules to the category \({}_k\mathcal M\) of left \(k\)-modules lifts the closed structure (although not necessarily the monoidal one) of \({}_k\mathcal M\), that is, such that which \(\omega\) is a strict monoidal functor. The main result of the paper gives necessary and sufficient conditions on the algebra \(A\) in order that the closed structure of \({}_k\mathcal M\) lifts to a skew-closed structure on \({}_A\mathcal M\); an algebra satisfying these conditions is called a \textit{gabi algebra}, apparently making reference to the terms ``\textit{generalized associative bialgebra}'', and/or to a private communication with Gabriella Böhm, where the conditions appeared for the first time. A gabi algebra \(A\) is called \textit{normal} if its category of modules is (associative and unital normal) closed with closed forgetful functor to \({}_k\mathcal M\).\N\NNecessary and sufficient conditions for a gabi-algebra \(A\) to be a (one-sided) Hopf algebra are given and it is shown that there is a bijective correspondence between normal gabi-algebra structures and Hopf algebra structures on \(A\).