Grothendieck-Verdier duality patterns in quantum algebra

Abstract: After a brief survey of the basic definitions of the Grothendieck--Verdier categories and dualities, I consider in this context introduced earlier dualities in the categories of quadratic algebras and operads, largely motivated by the theory of quantum groups. Finally, I argue that Dubrovin's "almost duality" in the theory of Frobenius manifolds and quantum cohomology also must fit a (possibly extended) version of Grothendieck--Verdier duality.