Opposite brace triples, Hopf braces and matched pairs of Hopf algebras

TL;DR

This paper introduces the category of opposite brace triples in a braided monoidal setting and establishes isomorphisms with Hopf braces and matched pairs of Hopf algebras under certain conditions.

Contribution

It generalizes the concept of Hopf braces by defining opposite brace triples and proves their equivalence to Hopf braces and matched pairs in specific cases.

Findings

Category of opposite brace triples is isomorphic to Hopf braces under cocommutativity.

Subcategories with fixed Hopf algebra are isomorphic to matched pairs over that algebra.

Provides a unified framework connecting braces and Hopf algebra structures.

Abstract

In this paper the category of opposite brace triples is introduced in a general braided monoidal setting. Under cocommutativity, it is proved to be isomorphic to the category of Hopf braces. Furthermore, if one considers the subcategories arising from fixing one of the underlying Hopf algebras, then these two categories are also isomorphic to the category of matched pairs over that Hopf algebra.