Grothendieck-Verdier duality patterns in quantum algebra

TL;DR

This paper explores how Grothendieck--Verdier duality patterns manifest in quantum algebra, connecting dualities in quadratic algebras, operads, and Frobenius manifolds within a unified framework.

Contribution

It introduces a unified perspective on Grothendieck--Verdier duality across various structures in quantum algebra and related fields.

Findings

Dualities in quadratic algebras and operads are compatible with Grothendieck--Verdier duality.

Dubrovin's 'almost duality' in Frobenius manifolds aligns with an extended Grothendieck--Verdier duality.

The paper provides a conceptual framework linking dualities in quantum groups and geometry.

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.