quantum heap

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.

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

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

satisfying the following properties

$(\id \otimes \id \otimes \tau) \tau
= (\tau \otimes \id \otimes \id) \tau$

$(\id \otimes \mu) \tau = \id \otimes 1_H$

$(\mu \otimes \id) \tau = 1_H \otimes \id$

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

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

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

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

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

- C. Grunspan,
*Quantum torsors*, math.QA/0204280

where the list has an additional unnecessary and superfluous axiom, later removed by

- P. Schauenburg,
*Quantum torsors and Hopf-Galois objects*, math.QA/0208047

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

- Tomasz Brzeziński, Joost Vercruysse,
*Bimodule herds*, arXiv:0805.2510

See also

- Thomas Booker, Ross Street,
*Torsors, herds and flocks*, arxiv/0912.4551 doi

Revised on November 30, 2012 21:04:34
by Zoran Škoda
(193.51.104.65)