Quantum increasing sequences generate quantum permutation groups

TL;DR

This paper proves the inner faithfulness of a map extending quantum increasing sequences to quantum permutation groups, using inductive techniques and recent advances in quantum group theory.

Contribution

It provides a general proof of the inner faithfulness of Curran's map, answering a question posed in 2016, by leveraging novel inductive methods and results on quantum permutation groups.

Findings

Inner faithfulness of Curran's map established in full generality.

Inductive approach reduces the problem to smaller algebras.

Utilizes recent results on quantum permutation groups.

Abstract

We answer a question of A. Skalski and P.M. So{\l}tan (2016) about inner faithfulness of the S.~Curran's map of extending a quantum increasing sequence to a quantum permutation in full generality. To do so, we exploit some novel techniques introduced by Banica (2018) and Brannan, Chirvasitu, Freslon (2018) concerned with the Banica's conjecture regarding quantum permutation groups. Roughly speaking, we find a inductive setting in which the inner faithfulness of Curran's map can be boiled down to inner faithfulness of similar map for smaller algebras and then rely on inductive generation result for quantum permutation groups of Brannan, Chirvasitu and Freslon.