# nLab quantum heap

## Idea

A quantum heap is a dual version of a notion of a heap internalized to the monoidal category $k$-Vect, with usual tensor product over $k$. A quantum heap is to a Hopf algebra what a heap is to a group.

## Definition

A quantum heap is an associative unital algebra $\left(H,\mu ,\eta \right)$ together with a ternary algebra cooperation

(1)$\tau :H\to H\otimes {H}_{\mathrm{op}}\otimes H,$\tau : H \rightarrow H \otimes H_{op} \otimes H,

satisfying the following properties

$\left(id\otimes id\otimes \tau \right)\tau =\left(\tau \otimes id\otimes id\right)\tau$(\id \otimes \id \otimes \tau) \tau = (\tau \otimes \id \otimes \id) \tau
$\left(id\otimes \mu \right)\tau =id\otimes {1}_{H}$(\id \otimes \mu) \tau = \id \otimes 1_H
$\left(\mu \otimes id\right)\tau ={1}_{H}\otimes id$(\mu \otimes \id) \tau = 1_H \otimes \id

Moreover, $\tau$ is required to be algebra homomorphism from $H$ into $H\otimes {H}_{\mathrm{op}}\otimes H$, where ${H}_{\mathrm{op}}$ has the opposite algebra structure and the tensor product has the usual algebra structure. We use heap analogue of the Sweedler notation:

(2)$\tau \left(h\right)=\sum {h}^{\left(1\right)}\otimes {h}^{\left(2\right)}\otimes {h}^{\left(3\right)},$\tau(h) = \sum h^{(1)} \otimes h^{(2)} \otimes h^{(3)},

and because of the first of the above identities, it is justified to extend it to any odd number $\ge 3$ factors, e.g.

(3)$\left(id\otimes id\otimes \tau \right)\tau \left(h\right)=\sum {h}^{\left(1\right)}\otimes {h}^{\left(2\right)}\otimes {h}^{\left(3\right)}\otimes {h}^{\left(4\right)}\otimes {h}^{\left(5\right)}.$(\id \otimes \id \otimes \tau) \tau(h) = \sum h^{(1)} \otimes h^{(2)} \otimes h^{(3)} \otimes h^{(4)} \otimes h^{(5)}.

## Versions and references

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

• M. Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 1 (1999) 35–72

The noncommutative case has been first introduced and studied in chapter 9 of

• Zoran Škoda, Cosets for quantum groups, Ph. D. thesis, University of Wisconsin, defended January 17, 2002

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

• Z. Škoda, Quantum heaps, cops and heapy categories, Mathematical Communications 12, No. 1, pp. 1–9 (2007) math.QA/0701749

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 $k$-algebra $A$ instead over $k$) 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

• G. Böhm, T. Brzeziński, Pre-torsors and equivalences, J. Algebra 317 544–580 (2007) (math.QA/0607529), Corrigendum: J Algebra 319 1339-1340 (2008)

and generalizations with emphasis on noncommutative base are studied in