Closed categories, modules and (one-sided) Hopf algebras

Abstract: A well-known characterization of Hopf algebras, that I always found fascinating and elegant, states that an algebra A over a field k is a Hopf algebra if and only if its category of modules is a closed monoidal category in such a way that the forgetful functor to vector spaces preserves the closed monoidal structure. We usually split this result into two steps: the lifting of the monoidal structure corresponds to the bialgebra structure, and then the further lifting of the closed structure as adjoint to the monoidal one corresponds to the existence of an antipode. However, closed structures can be defined independently of monoidal ones and have their own dignity and importance. Which new structure on our algebra A would correspond to lifting the closed structure of vector spaces alone? How would this relate with the familiar bialgebra and Hopf algebra structures coming from lifting the monoidal and closed monoidal ones? It turns out that lifting the closed structure corresponds to the existence of algebra maps 𝛿 : A -> A⊗A^op and ε : A -> k satisfying appropriate conditions. Moreover, a quite unexpected source of examples is provided by certain one-sided Hopf algebras, i.e. bialgebras with a morphism which is just a one-sided convolution inverse of the identity. In this seminar, based on an ongoing collaboration with Johannes Berger and Joost Vercruysse which is continuing discussions with Gabriella Böhm, I will present our progresses in the study of these new algebraic structures.

Mathematics

Audience: researchers in the topic

European Quantum Algebra Lectures (EQuAL)

Series comments: EQuAL is an online seminar series on quantum algebra and related topics such as topological and conformal field theory, operator algebra, representation theory, quantum topology, etc. sites.google.com/view/equalseminar/home