A quantum heap is an associative unital algebra together with a ternary algebra cooperation
satisfying the following properties
Moreover, is required to be algebra homomorphism from into , where has the opposite algebra structure and the tensor product has the usual algebra structure. We use heap analogue of the Sweedler notation:
and because of the first of the above identities, it is justified to extend it to any odd number factors, e.g.
The algebra of functions on an affine variety with a structure of an algebraic heap is an example of a commutative heap. This has been used explicitly in
The noncommutative case has been first introduced and studied in chapter 9 of
Choosing a character of a quantum heap one obtains a copointed quantum heap. According to the main theorem in that chapter, the category of copointed quantum heap is equivalent to the category of Hopf algebras. An updated version of that chapter 9 has been published only much later in
where also a notion of “heapy category”, sort of monoidal-category like categorification of a heap is proposed.
Independently, a relative version (that is over commutative -algebra instead over ) of a quantum heap has been proposed in 2002 under the name “quantum torsor” by
where the list has an additional unnecessary and superfluous axiom, later removed by
An interesting notion in Grunspan’s work are the left and right automorphism Hopf algebras of a quantum heap which are analogues of the automorphism group of a heap; in addition he has exhibited a counterexample showing that the left and right automorphism Hopf algebra do not need to be isomorphic.
A recent related article is
and generalizations with emphasis on noncommutative base are studied in