Geodesic
Grothendieck--Verdier duality patterns in quantum algebra
Izvestiya. Mathematics , Tome 81 (2017) no. 4, pp. 818-826

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After a brief survey of the basic definitions of Grothendieck–Verdier categories and dualities, I consider in this context dualities introduced earlier 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 must also fit a (possibly extended) version of Grothendieck–Verdier duality.
Keywords: duality, $F$-manifolds, quadratic algebras, quadratic operads. 
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     author = {Yu. I. Manin},
     title = {Grothendieck--Verdier duality patterns in quantum algebra},
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Yu. I. Manin. Grothendieck--Verdier duality patterns in quantum algebra. Izvestiya. Mathematics , Tome 81 (2017) no. 4, pp. 818-826. http://geodesic.mathdoc.fr/item/IM2_2017_81_4_a4/