Paolo Saracco

A classical result in Tannaka-Krein reconstruction theory states that there is a bijective correspondence between 

(a) Hopf algebra structures on an algebra A over a commutative ring \Bbbk 

and 

(b) closed monoidal structures on the category of left A-modules such that the forgetful functor to the category of \Bbbk-modules is closed monoidal.

One would usually split this result into two steps: the lifting of the monoidal structure corresponds to the bialgebra structure, and then the lifting of the closed structure (as adjoint to the monoidal one) corresponds to the existence of an antipode.

It is less known, however, that closed structures on a category can be defined independently of monoidal ones and this led us to explore a new perspective on Tannaka-Krein reconstruction: what kind of structure on A would correspond to lifting the closed structure of \Bbbk-modules alone?

In this talk we will see how lifting the closed structure corresponds to the existence of a gabi-algebra structure on A, i.e. a pair of algebra maps \delta: A -> A \otimes A^{op}$ and \varepsilon: A -> \Bbbk satisfying appropriate conditions. 

Additionally, we will see that the lifted closed structure on the category of A-modules is normal if and only if A is a Hopf algebra, thus supporting our claim that everybody knows what a normal gabi-algebra is.