A topological origin of quantum symmetric pairs

TL;DR

This paper introduces involutive little disks operads as Z/2Z-orbifold versions of the classical operads, classifies their categorical algebras, and connects them to quantum symmetric pairs via topological and categorical structures.

Contribution

It provides a classification of involutive little disks operad algebras and links them to quantum symmetric pairs through topological and categorical frameworks.

Findings

Categorical algebras correspond to braided monoidal categories with anti-involution.

Explicit classification via functors, isomorphisms, and coherence equations.

Main examples include categories of representations of quantum groups with involution.

Abstract

It is well known that braided monoidal categories are the categorical algebras of the little two-dimensional disks operad. We introduce involutive little disks operads, which are Z/2Z-orbifold versions of the little disks operads. We classify their categorical algebras and describe these explicitly in terms of a finite list of functors, natural isomorphisms and coherence equations. In dimension two, the categorical algebras are braided monoidal categories with an anti-involution together with a pointed module category carrying a universal solution to the (twisted) reflection equation. Main examples are obtained from the categories of representations of a ribbon Hopf algebra with an involution and a quasi-triangular coideal subalgebra, such as a quantum group and a quantum symmetric pair coideal subalgebra.