Quantum heaps, cops and heapy categories

TL;DR

This paper introduces quantum heaps as associative algebras with ternary operations, establishing their relation to Hopf algebras and defining a new category called heapy categories based on their representations.

Contribution

It defines quantum heaps and heapy categories, extending heap theory to quantum algebra and relating these structures to Hopf algebras and their representations.

Findings

Quantum heaps are associative algebras with ternary cooperation.

The category of copointed quantum heaps is isomorphic to Hopf algebras.

Representations of quantum heaps form a new kind of monoidal category called heapy categories.

Abstract

A heap is a structure with a ternary operation which is intuitively a group with forgotten unit element. Quantum heaps are associative algebras with a ternary cooperation which are to the Hopf algebras what heaps are to groups, and, in particular, the category of copointed quantum heaps is isomorphic to the category of Hopf algebras. There is an intermediate structure of a cop in monoidal category which is in the case of vector spaces to a quantum heap about what is a coalgebra to a Hopf algebra. The representations of Hopf algebras make a rigid monoidal category. Similarly the representations of quantum heaps make a kind of category with ternary products, which we call a heapy category.